Taxicab numbers

From Rosetta Code
Task
Taxicab numbers
You are encouraged to solve this task according to the task description, using any language you may know.

A   taxicab number   (the definition that is being used here)   is a positive integer that can be expressed as the sum of two positive cubes in more than one way.


The first taxicab number is   1729,   which is:

13   +   123       and also
93   +   103.


Taxicab numbers are also known as:

  •   taxi numbers
  •   taxi-cab numbers
  •   taxi cab numbers
  •   Hardy-Ramanujan numbers


Task
  • Compute and display the lowest 25 taxicab numbers (in numeric order, and in a human-readable format).
  • For each of the taxicab numbers, show the number as well as it's constituent cubes.


Extra credit
  • Show the 2,000th taxicab number, and a half dozen more


See also



11l

Translation of: Python
V cubes = (1..1199).map(x -> Int64(x) ^ 3)
[Int64 = Int64] crev
L(x3) cubes
   crev[x3] = L.index + 1

V sums = sorted(multiloop_filtered(cubes, cubes, (x, y) -> y < x, (x, y) -> x + y))

V idx = 0
L(i) 1 .< sums.len - 1
   I sums[i - 1] != sums[i] & sums[i] == sums[i + 1]
      idx++
      I (idx > 25 & idx < 2000) | idx > 2006
         L.continue

      V n = sums[i]
      [(Int64, Int64)] p
      L(x) cubes
         I n - x < x
            L.break
         I n - x C crev
            p.append((crev[x], crev[n - x]))
      print(‘#4: #10’.format(idx, n), end' ‘ ’)
      L(x1, x2) p
         print(‘ = #4^3 + #4^3’.format(x1, x2), end' ‘ ’)
      print()
Output:
   1:       1729  =    1^3 +   12^3  =    9^3 +   10^3
   2:       4104  =    2^3 +   16^3  =    9^3 +   15^3
   3:      13832  =    2^3 +   24^3  =   18^3 +   20^3
   4:      20683  =   10^3 +   27^3  =   19^3 +   24^3
   5:      32832  =    4^3 +   32^3  =   18^3 +   30^3
   6:      39312  =    2^3 +   34^3  =   15^3 +   33^3
   7:      40033  =    9^3 +   34^3  =   16^3 +   33^3
   8:      46683  =    3^3 +   36^3  =   27^3 +   30^3
   9:      64232  =   17^3 +   39^3  =   26^3 +   36^3
  10:      65728  =   12^3 +   40^3  =   31^3 +   33^3
  11:     110656  =    4^3 +   48^3  =   36^3 +   40^3
  12:     110808  =    6^3 +   48^3  =   27^3 +   45^3
  13:     134379  =   12^3 +   51^3  =   38^3 +   43^3
  14:     149389  =    8^3 +   53^3  =   29^3 +   50^3
  15:     165464  =   20^3 +   54^3  =   38^3 +   48^3
  16:     171288  =   17^3 +   55^3  =   24^3 +   54^3
  17:     195841  =    9^3 +   58^3  =   22^3 +   57^3
  18:     216027  =    3^3 +   60^3  =   22^3 +   59^3
  19:     216125  =    5^3 +   60^3  =   45^3 +   50^3
  20:     262656  =    8^3 +   64^3  =   36^3 +   60^3
  21:     314496  =    4^3 +   68^3  =   30^3 +   66^3
  22:     320264  =   18^3 +   68^3  =   32^3 +   66^3
  23:     327763  =   30^3 +   67^3  =   51^3 +   58^3
  24:     373464  =    6^3 +   72^3  =   54^3 +   60^3
  25:     402597  =   42^3 +   69^3  =   56^3 +   61^3
2000: 1671816384  =  428^3 + 1168^3  =  940^3 +  944^3
2001: 1672470592  =   29^3 + 1187^3  =  632^3 + 1124^3
2002: 1673170856  =  458^3 + 1164^3  =  828^3 + 1034^3
2003: 1675045225  =  522^3 + 1153^3  =  744^3 + 1081^3
2004: 1675958167  =  492^3 + 1159^3  =  711^3 + 1096^3
2005: 1676926719  =   63^3 + 1188^3  =  714^3 + 1095^3
2006: 1677646971  =   99^3 + 1188^3  =  891^3 +  990^3

AWK

# syntax: GAWK -f TAXICAB_NUMBERS.AWK
BEGIN {
    stop = 99
    for (a=1; a<=stop; a++) {
      for (b=1; b<=stop; b++) {
        n1 = a^3 + b^3
        for (c=1; c<=stop; c++) {
          if (a == c) { continue }
          for (d=1; d<=stop; d++) {
            n2 = c^3 + d^3
            if (n1 == n2 && (a != d || b != c)) {
              if (n1 in arr) { continue }
              arr[n1] = sprintf("%7d = %2d^3 + %2d^3 = %2d^3 + %2d^3",n1,a,b,c,d)
            }
          }
        }
      }
    }
    PROCINFO["sorted_in"] = "@ind_num_asc"
    for (i in arr) {
      if (++count <= 25) {
        printf("%2d: %s\n",count,arr[i])
      }
    }
    printf("\nThere are %d taxicab numbers using bounds of %d\n",length(arr),stop)
    exit(0)
}
Output:
 1:    1729 =  1^3 + 12^3 =  9^3 + 10^3
 2:    4104 =  2^3 + 16^3 =  9^3 + 15^3
 3:   13832 =  2^3 + 24^3 = 18^3 + 20^3
 4:   20683 = 10^3 + 27^3 = 19^3 + 24^3
 5:   32832 =  4^3 + 32^3 = 18^3 + 30^3
 6:   39312 =  2^3 + 34^3 = 15^3 + 33^3
 7:   40033 =  9^3 + 34^3 = 16^3 + 33^3
 8:   46683 =  3^3 + 36^3 = 27^3 + 30^3
 9:   64232 = 17^3 + 39^3 = 26^3 + 36^3
10:   65728 = 12^3 + 40^3 = 31^3 + 33^3
11:  110656 =  4^3 + 48^3 = 36^3 + 40^3
12:  110808 =  6^3 + 48^3 = 27^3 + 45^3
13:  134379 = 12^3 + 51^3 = 38^3 + 43^3
14:  149389 =  8^3 + 53^3 = 29^3 + 50^3
15:  165464 = 20^3 + 54^3 = 38^3 + 48^3
16:  171288 = 17^3 + 55^3 = 24^3 + 54^3
17:  195841 =  9^3 + 58^3 = 22^3 + 57^3
18:  216027 =  3^3 + 60^3 = 22^3 + 59^3
19:  216125 =  5^3 + 60^3 = 45^3 + 50^3
20:  262656 =  8^3 + 64^3 = 36^3 + 60^3
21:  314496 =  4^3 + 68^3 = 30^3 + 66^3
22:  320264 = 18^3 + 68^3 = 32^3 + 66^3
23:  327763 = 30^3 + 67^3 = 51^3 + 58^3
24:  373464 =  6^3 + 72^3 = 54^3 + 60^3
25:  402597 = 42^3 + 69^3 = 56^3 + 61^3
                                                 
There are 45 taxicab numbers using bounds of 99  

Befunge

This is quite slow in most interpreters, although a decent compiler should allow it to complete in a matter of seconds. Regardless of the speed, though, the range in a standard Befunge-93 implementation is limited to the first 64 numbers in the series, after which the 8-bit memory cells will overflow. That range could be extended in Befunge-98, but realistically you're not likely to wait that long for the results.

v+1$$<_v#!`**::+1g42$$_v#<!`**::+1g43\g43::<<v,,.g42,<
>004p:0>1+24p:24g\:24g>>1+:34p::**24g::**+-|p>9,,,14v,
,,,"^3 + ^3= ^3 + ^3".\,,,9"= ".:\_v#g40g43<^v,,,,.g<^
5+,$$$\1+:38*`#@_\::"~"1+:24p34p0\0>14p24g04^>,04g.,,5
Output:
1729    = 10 ^3 + 9 ^3  = 12 ^3 + 1 ^3
4104    = 15 ^3 + 9 ^3  = 16 ^3 + 2 ^3
13832   = 20 ^3 + 18 ^3 = 24 ^3 + 2 ^3
20683   = 24 ^3 + 19 ^3 = 27 ^3 + 10 ^3
32832   = 30 ^3 + 18 ^3 = 32 ^3 + 4 ^3
39312   = 33 ^3 + 15 ^3 = 34 ^3 + 2 ^3
40033   = 33 ^3 + 16 ^3 = 34 ^3 + 9 ^3
46683   = 30 ^3 + 27 ^3 = 36 ^3 + 3 ^3
64232   = 36 ^3 + 26 ^3 = 39 ^3 + 17 ^3
65728   = 33 ^3 + 31 ^3 = 40 ^3 + 12 ^3
110656  = 40 ^3 + 36 ^3 = 48 ^3 + 4 ^3
110808  = 45 ^3 + 27 ^3 = 48 ^3 + 6 ^3
134379  = 43 ^3 + 38 ^3 = 51 ^3 + 12 ^3
149389  = 50 ^3 + 29 ^3 = 53 ^3 + 8 ^3
165464  = 48 ^3 + 38 ^3 = 54 ^3 + 20 ^3
171288  = 54 ^3 + 24 ^3 = 55 ^3 + 17 ^3
195841  = 57 ^3 + 22 ^3 = 58 ^3 + 9 ^3
216027  = 59 ^3 + 22 ^3 = 60 ^3 + 3 ^3
216125  = 50 ^3 + 45 ^3 = 60 ^3 + 5 ^3
262656  = 60 ^3 + 36 ^3 = 64 ^3 + 8 ^3
314496  = 66 ^3 + 30 ^3 = 68 ^3 + 4 ^3
320264  = 66 ^3 + 32 ^3 = 68 ^3 + 18 ^3
327763  = 58 ^3 + 51 ^3 = 67 ^3 + 30 ^3
373464  = 60 ^3 + 54 ^3 = 72 ^3 + 6 ^3
402597  = 61 ^3 + 56 ^3 = 69 ^3 + 42 ^3

C

Using a priority queue to emit sum of two cubs in order. It's reasonably fast and doesn't use excessive amount of memory (the heap is only at 245 length upon the 2006th taxi).

#include <stdio.h>
#include <stdlib.h>

typedef unsigned long long xint;
typedef unsigned uint;
typedef struct {
	uint x, y; // x > y always
	xint value;
} sum_t;

xint *cube;
uint n_cubes;

sum_t *pq;
uint pq_len, pq_cap;

void add_cube(void)
{
	uint x = n_cubes++;
	cube = realloc(cube, sizeof(xint) * (n_cubes + 1));
	cube[n_cubes] = (xint) n_cubes*n_cubes*n_cubes;
	if (x < 2) return; // x = 0 or 1 is useless

	if (++pq_len >= pq_cap) {
		if (!(pq_cap *= 2)) pq_cap = 2;
		pq = realloc(pq, sizeof(*pq) * pq_cap);
	}

	sum_t tmp = (sum_t) { x, 1, cube[x] + 1 };
	// upheap
	uint i, j;
	for (i = pq_len; i >= 1 && pq[j = i>>1].value > tmp.value; i = j)
		pq[i] = pq[j];

	pq[i] = tmp;
}

void next_sum(void)
{
redo:	while (!pq_len || pq[1].value >= cube[n_cubes]) add_cube();

	sum_t tmp = pq[0] = pq[1];	// pq[0] always stores last seen value
	if (++tmp.y >= tmp.x) {		// done with this x; throw it away
		tmp = pq[pq_len--];
		if (!pq_len) goto redo;	// refill empty heap
	} else
		tmp.value += cube[tmp.y] - cube[tmp.y-1];

	uint i, j;
	// downheap
	for (i = 1; (j = i<<1) <= pq_len; pq[i] = pq[j], i = j) {
		if (j < pq_len && pq[j+1].value < pq[j].value) ++j;
		if (pq[j].value >= tmp.value) break;
	}
	pq[i] = tmp;
}

uint next_taxi(sum_t *hist)
{
	do next_sum(); while (pq[0].value != pq[1].value);

	uint len = 1;
	hist[0] = pq[0];
	do {
		hist[len++] = pq[1]; 
		next_sum();
	} while (pq[0].value == pq[1].value);

	return len;
}

int main(void)
{
	uint i, l;
	sum_t x[10];
	for (i = 1; i <= 2006; i++) {
		l = next_taxi(x);
		if (25 < i && i < 2000) continue;
		printf("%4u:%10llu", i, x[0].value);
		while (l--) printf(" = %4u^3 + %4u^3", x[l].x, x[l].y);
		putchar('\n');
	}
	return 0;
}
Output:
   1:      1729 =   12^3 +    1^3 =   10^3 +    9^3
   2:      4104 =   15^3 +    9^3 =   16^3 +    2^3
   3:     13832 =   20^3 +   18^3 =   24^3 +    2^3
   4:     20683 =   27^3 +   10^3 =   24^3 +   19^3
   5:     32832 =   30^3 +   18^3 =   32^3 +    4^3
   6:     39312 =   33^3 +   15^3 =   34^3 +    2^3
   7:     40033 =   33^3 +   16^3 =   34^3 +    9^3
   8:     46683 =   30^3 +   27^3 =   36^3 +    3^3
   9:     64232 =   36^3 +   26^3 =   39^3 +   17^3
  10:     65728 =   33^3 +   31^3 =   40^3 +   12^3
  11:    110656 =   40^3 +   36^3 =   48^3 +    4^3
  12:    110808 =   45^3 +   27^3 =   48^3 +    6^3
  13:    134379 =   43^3 +   38^3 =   51^3 +   12^3
  14:    149389 =   50^3 +   29^3 =   53^3 +    8^3
  15:    165464 =   48^3 +   38^3 =   54^3 +   20^3
  16:    171288 =   54^3 +   24^3 =   55^3 +   17^3
  17:    195841 =   57^3 +   22^3 =   58^3 +    9^3
  18:    216027 =   59^3 +   22^3 =   60^3 +    3^3
  19:    216125 =   50^3 +   45^3 =   60^3 +    5^3
  20:    262656 =   60^3 +   36^3 =   64^3 +    8^3
  21:    314496 =   66^3 +   30^3 =   68^3 +    4^3
  22:    320264 =   66^3 +   32^3 =   68^3 +   18^3
  23:    327763 =   58^3 +   51^3 =   67^3 +   30^3
  24:    373464 =   60^3 +   54^3 =   72^3 +    6^3
  25:    402597 =   61^3 +   56^3 =   69^3 +   42^3
2000:1671816384 = 1168^3 +  428^3 =  944^3 +  940^3
2001:1672470592 = 1124^3 +  632^3 = 1187^3 +   29^3
2002:1673170856 = 1034^3 +  828^3 = 1164^3 +  458^3
2003:1675045225 = 1153^3 +  522^3 = 1081^3 +  744^3
2004:1675958167 = 1096^3 +  711^3 = 1159^3 +  492^3
2005:1676926719 = 1188^3 +   63^3 = 1095^3 +  714^3
2006:1677646971 =  990^3 +  891^3 = 1188^3 +   99^3

C++

Translation of: C#
#include <algorithm>
#include <iomanip>
#include <iostream>
#include <map>
#include <sstream>
#include <vector>

template <typename T>
size_t indexOf(const std::vector<T> &v, const T &k) {
    auto it = std::find(v.cbegin(), v.cend(), k);

    if (it != v.cend()) {
        return it - v.cbegin();
    }
    return -1;
}

int main() {
    std::vector<size_t> cubes;

    auto dump = [&cubes](const std::string &title, const std::map<int, size_t> &items) {
        std::cout << title;
        for (auto &item : items) {
            std::cout << "\n" << std::setw(4) << item.first << " " << std::setw(10) << item.second;
            for (auto x : cubes) {
                auto y = item.second - x;
                if (y < x) {
                    break;
                }
                if (std::count(cubes.begin(), cubes.end(), y)) {
                    std::cout << " = " << std::setw(4) << indexOf(cubes, y) << "^3 + " << std::setw(3) << indexOf(cubes, x) << "^3";
                }
            }
        }
    };

    std::vector<size_t> sums;

    // create sorted list of cube sums
    for (size_t i = 0; i < 1190; i++) {
        auto cube = i * i * i;
        cubes.push_back(cube);
        for (auto j : cubes) {
            sums.push_back(cube + j);
        }
    }
    std::sort(sums.begin(), sums.end());

    // now seek consecutive sums that match
    auto nm1 = sums[0];
    auto n = sums[1];
    int idx = 0;
    std::map<int, size_t> task;
    std::map<int, size_t> trips;

    auto it = sums.cbegin();
    auto end = sums.cend();
    it++;
    it++;

    while (it != end) {
        auto np1 = *it;

        if (nm1 == np1) {
            trips.emplace(idx, n);
        }
        if (nm1 != n && n == np1) {
            if (++idx <= 25 || idx >= 2000 == idx <= 2006) {
                task.emplace(idx, n);
            }
        }
        nm1 = n;
        n = np1;

        it++;
    }

    dump("First 25 Taxicab Numbers, the 2000th, plus the next half-dozen:", task);

    std::stringstream ss;
    ss << "\n\nFound " << trips.size() << " triple Taxicabs under 2007:";
    dump(ss.str(), trips);

    return 0;
}
Output:
First 25 Taxicab Numbers, the 2000th, plus the next half-dozen:
   1       1729 =   12^3 +   1^3 =   10^3 +   9^3
   2       4104 =   16^3 +   2^3 =   15^3 +   9^3
   3      13832 =   24^3 +   2^3 =   20^3 +  18^3
   4      20683 =   27^3 +  10^3 =   24^3 +  19^3
   5      32832 =   32^3 +   4^3 =   30^3 +  18^3
   6      39312 =   34^3 +   2^3 =   33^3 +  15^3
   7      40033 =   34^3 +   9^3 =   33^3 +  16^3
   8      46683 =   36^3 +   3^3 =   30^3 +  27^3
   9      64232 =   39^3 +  17^3 =   36^3 +  26^3
  10      65728 =   40^3 +  12^3 =   33^3 +  31^3
  11     110656 =   48^3 +   4^3 =   40^3 +  36^3
  12     110808 =   48^3 +   6^3 =   45^3 +  27^3
  13     134379 =   51^3 +  12^3 =   43^3 +  38^3
  14     149389 =   53^3 +   8^3 =   50^3 +  29^3
  15     165464 =   54^3 +  20^3 =   48^3 +  38^3
  16     171288 =   55^3 +  17^3 =   54^3 +  24^3
  17     195841 =   58^3 +   9^3 =   57^3 +  22^3
  18     216027 =   60^3 +   3^3 =   59^3 +  22^3
  19     216125 =   60^3 +   5^3 =   50^3 +  45^3
  20     262656 =   64^3 +   8^3 =   60^3 +  36^3
  21     314496 =   68^3 +   4^3 =   66^3 +  30^3
  22     320264 =   68^3 +  18^3 =   66^3 +  32^3
  23     327763 =   67^3 +  30^3 =   58^3 +  51^3
  24     373464 =   72^3 +   6^3 =   60^3 +  54^3
  25     402597 =   69^3 +  42^3 =   61^3 +  56^3
2000 1671816384 = 1168^3 + 428^3 =  944^3 + 940^3
2001 1672470592 = 1187^3 +  29^3 = 1124^3 + 632^3
2002 1673170856 = 1164^3 + 458^3 = 1034^3 + 828^3
2003 1675045225 = 1153^3 + 522^3 = 1081^3 + 744^3
2004 1675958167 = 1159^3 + 492^3 = 1096^3 + 711^3
2005 1676926719 = 1188^3 +  63^3 = 1095^3 + 714^3
2006 1677646971 = 1188^3 +  99^3 =  990^3 + 891^3

Found 10 triple Taxicabs under 2007:
 455   87539319 =  436^3 + 167^3 =  423^3 + 228^3 =  414^3 + 255^3
 535  119824488 =  493^3 +  11^3 =  492^3 +  90^3 =  428^3 + 346^3
 588  143604279 =  522^3 + 111^3 =  460^3 + 359^3 =  423^3 + 408^3
 655  175959000 =  560^3 +  70^3 =  552^3 + 198^3 =  525^3 + 315^3
 888  327763000 =  670^3 + 300^3 =  661^3 + 339^3 =  580^3 + 510^3
1299  700314552 =  872^3 + 334^3 =  846^3 + 456^3 =  828^3 + 510^3
1398  804360375 =  930^3 +  15^3 =  927^3 + 198^3 =  920^3 + 295^3
1515  958595904 =  986^3 +  22^3 =  984^3 + 180^3 =  856^3 + 692^3
1660 1148834232 = 1044^3 + 222^3 =  920^3 + 718^3 =  846^3 + 816^3
1837 1407672000 = 1120^3 + 140^3 = 1104^3 + 396^3 = 1050^3 + 630^3

C#

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;

namespace TaxicabNumber
{
    class Program
    {
        static void Main(string[] args)
        {
            IDictionary<long, IList<Tuple<int, int>>> taxicabNumbers = GetTaxicabNumbers(2006);
            PrintTaxicabNumbers(taxicabNumbers);
            Console.ReadKey();
        }

        private static IDictionary<long, IList<Tuple<int, int>>> GetTaxicabNumbers(int length)
        {
            SortedList<long, IList<Tuple<int, int>>> sumsOfTwoCubes = new SortedList<long, IList<Tuple<int, int>>>();

            for (int i = 1; i < int.MaxValue; i++)
            {
                for (int j = 1; j < int.MaxValue; j++)
                {
                    long sum = (long)(Math.Pow((double)i, 3) + Math.Pow((double)j, 3));

                    if (!sumsOfTwoCubes.ContainsKey(sum))
                    {
                        sumsOfTwoCubes.Add(sum, new List<Tuple<int, int>>());
                    }

                    sumsOfTwoCubes[sum].Add(new Tuple<int, int>(i, j));

                    if (j >= i)
                    {
                        break;
                    }
                }

                // Found that you need to keep going for a while after the length, because higher i values fill in gaps
                if (sumsOfTwoCubes.Count(t => t.Value.Count >= 2) >= length * 1.1)
                {
                    break;
                }
            }

            IDictionary<long, IList<Tuple<int, int>>> values = (from t in sumsOfTwoCubes where t.Value.Count >= 2 select t)
                .Take(2006)
                .ToDictionary(u => u.Key, u => u.Value);

            return values;
        }

        private static void PrintTaxicabNumbers(IDictionary<long, IList<Tuple<int, int>>> values)
        {
            int i = 1;

            foreach (long taxicabNumber in values.Keys)
            {
                StringBuilder output = new StringBuilder().AppendFormat("{0,10}\t{1,4}", i, taxicabNumber);

                foreach (Tuple<int, int> numbers in values[taxicabNumber])
                {
                    output.AppendFormat("\t= {0}^3 + {1}^3", numbers.Item1, numbers.Item2);
                }

                if (i <= 25 || (i >= 2000 && i <= 2006))
                {
                    Console.WriteLine(output.ToString());
                }

                i++;
            }
        }
    }
}

Alternate Algorithm

Based on the second Python example where only the sums are stored and sorted. Also shows the first 10 Taxicab Number triples.

using System; using static System.Console;
using System.Collections.Generic; using System.Linq;

class Program {

  static void Main(string[] args) {

    List<uint> cubes = new List<uint>(), sums = new List<uint>();

    void dump(string title, Dictionary <int, uint> items) {
      Write(title); foreach (var item in items) {
        Write("\n{0,4} {1,10}", item.Key, item.Value);
        foreach (uint x in cubes) { uint y = item.Value - x;
          if (y < x) break; if (cubes.Contains(y))
            Write(" = {0,4}³ + {1,3}³", cubes.IndexOf(y), cubes.IndexOf(x));
      } } }

    DateTime st = DateTime.Now; 
    // create sorted list of cube sums
    for (uint i = 0, cube; i < 1190; i++) { cube = i * i * i;
      cubes.Add(cube); foreach (uint j in cubes)
        sums.Add(cube + j); } sums.Sort();
    // now seek consecutive sums that match
    uint nm1 = sums[0], n = sums[1]; int idx = 0;
    Dictionary <int, uint> task = new Dictionary <int, uint>(),
                           trips = new Dictionary <int, uint>();
    foreach (var np1 in sums.Skip(2)) {
      if (nm1 == np1) trips.Add(idx, n); if (nm1 != n && n == np1)
        if (++idx <= 25 || idx >= 2000 == idx <= 2006)
          task.Add(idx, n); nm1 = n; n = np1; }
    // show results
    dump("First 25 Taxicab Numbers, the 2000th, plus the next half-dozen:", task);
    dump(string.Format("\n\nFound {0} triple Taxicabs under {1}:", trips.Count, 2007), trips);
    Write("\n\nElasped: {0}ms", (DateTime.Now - st).TotalMilliseconds); }
}
Output:

(from TIO.run)

First 25 Taxicab Numbers, the 2000th, plus the next half-dozen:
   1       1729 =   12³ +   1³ =   10³ +   9³
   2       4104 =   16³ +   2³ =   15³ +   9³
   3      13832 =   24³ +   2³ =   20³ +  18³
   4      20683 =   27³ +  10³ =   24³ +  19³
   5      32832 =   32³ +   4³ =   30³ +  18³
   6      39312 =   34³ +   2³ =   33³ +  15³
   7      40033 =   34³ +   9³ =   33³ +  16³
   8      46683 =   36³ +   3³ =   30³ +  27³
   9      64232 =   39³ +  17³ =   36³ +  26³
  10      65728 =   40³ +  12³ =   33³ +  31³
  11     110656 =   48³ +   4³ =   40³ +  36³
  12     110808 =   48³ +   6³ =   45³ +  27³
  13     134379 =   51³ +  12³ =   43³ +  38³
  14     149389 =   53³ +   8³ =   50³ +  29³
  15     165464 =   54³ +  20³ =   48³ +  38³
  16     171288 =   55³ +  17³ =   54³ +  24³
  17     195841 =   58³ +   9³ =   57³ +  22³
  18     216027 =   60³ +   3³ =   59³ +  22³
  19     216125 =   60³ +   5³ =   50³ +  45³
  20     262656 =   64³ +   8³ =   60³ +  36³
  21     314496 =   68³ +   4³ =   66³ +  30³
  22     320264 =   68³ +  18³ =   66³ +  32³
  23     327763 =   67³ +  30³ =   58³ +  51³
  24     373464 =   72³ +   6³ =   60³ +  54³
  25     402597 =   69³ +  42³ =   61³ +  56³
2000 1671816384 = 1168³ + 428³ =  944³ + 940³
2001 1672470592 = 1187³ +  29³ = 1124³ + 632³
2002 1673170856 = 1164³ + 458³ = 1034³ + 828³
2003 1675045225 = 1153³ + 522³ = 1081³ + 744³
2004 1675958167 = 1159³ + 492³ = 1096³ + 711³
2005 1676926719 = 1188³ +  63³ = 1095³ + 714³
2006 1677646971 = 1188³ +  99³ =  990³ + 891³

Found 10 triple Taxicabs under 2007:
 455   87539319 =  436³ + 167³ =  423³ + 228³ =  414³ + 255³
 535  119824488 =  493³ +  11³ =  492³ +  90³ =  428³ + 346³
 588  143604279 =  522³ + 111³ =  460³ + 359³ =  423³ + 408³
 655  175959000 =  560³ +  70³ =  552³ + 198³ =  525³ + 315³
 888  327763000 =  670³ + 300³ =  661³ + 339³ =  580³ + 510³
1299  700314552 =  872³ + 334³ =  846³ + 456³ =  828³ + 510³
1398  804360375 =  930³ +  15³ =  927³ + 198³ =  920³ + 295³
1515  958595904 =  986³ +  22³ =  984³ + 180³ =  856³ + 692³
1660 1148834232 = 1044³ + 222³ =  920³ + 718³ =  846³ + 816³
1837 1407672000 = 1120³ + 140³ = 1104³ + 396³ = 1050³ + 630³

Elasped: 78.7948ms

Clojure

(ns test-project-intellij.core
  (:gen-class))

(defn cube [x]
  "Cube a number through triple multiplication"
  (* x x x))

(defn sum3 [[i j]]
   " [i j] -> i^3 + j^3"
  (+ (cube i) (cube j)))

(defn next-pair [[i j]]
  " Generate next [i j] pair of sequence  (producing lower triangle pairs) "
  (if (< j i)
    [i (inc j)]
    [(inc i) 1]))

;; Pair sequence generator [1 1] [2 1] [2 2] [3 1] [3 2] [3 3] ...
(def pairs-seq	(iterate next-pair [1 1]))

(defn dict-inc [m pair]
  " Add pair to pair map m, with the key of the map based upon the cubic sum (sum3) and the value appends the pair "
  (update-in m [(sum3 pair)] (fnil #(conj % pair) [])))

(defn enough? [m n-to-generate]
  " Checks if we have enough taxi numbers (i.e. if number in map >= count-needed "
  (->> m                                ; hash-map of sum of cube of numbers [key] and their pairs as value
       (filter #(if (> (count (second %)) 1) true false))   ; filter out ones which don't have more than 1 entry
       (count)                                              ; count the item remaining
       (<= n-to-generate)))                                ; true iff count-needed is less or equal to the nubmer filtered

(defn find-taxi-numbers [n-to-generate]
  " Generates 1st n-to-generate taxi numbers"
  (loop [m {}               ; Hash-map containing cube of pairs (key) and set of pairs that produce sum (value)
         p pairs-seq        ; select pairs from our pair sequence generator (i.e. [1 1] [2 1] [2 2] ...)
         num-tried 0        ; Since its expensve to count how many taxi numbers we have found
         check-after 1]     ; we only check if we have enough numbers every time (num-tried equals check-after)
                            ; num-tried increments by 1 each time we try the next pair and
                            ; check-after doubles if we don't have enough taxi numbers
    (if (and (= num-tried check-after) (enough? m n-to-generate)) ; check if we found enough taxi numbers
      (sort-by first (into [] (filter #(> (count (second %)) 1) m)))  ; sort the taxi numbers and this is the result
      (if (= num-tried check-after)                                   ; Check if we need to increase our count between checking
        (recur (dict-inc m (first p)) (rest p) (inc num-tried) (* 2 check-after))   ; increased count between checking
        (recur (dict-inc m (first p)) (rest p) (inc num-tried) check-after)))))     ; didn't increase the count

; Generate 1st 2006 taxi numbers
(def result (find-taxi-numbers 2006))

;; Show First 25
(defn show-result [n sample]
  " Prints one line of result "
  (print (format "%4d:%10d" n  (first sample)))
  (doseq [q  (second sample)
          :let [[i j] q]]
      (print (format " = %4d^3 + %4d^3" i j)))
  (println))

; 1st 25 taxi numbers
(doseq [n (range 1 26)
        :let [sample (nth result (dec n))]]
  (show-result n sample))

; taxi numbers from 2000th to 2006th
(doseq [n (range 2000 2007)
        :let [sample (nth result (dec n))]]
  (show-result n sample))

}
Output:
  1:      1729 =   10^3 +    9^3 =   12^3 +    1^3
   2:      4104 =   15^3 +    9^3 =   16^3 +    2^3
   3:     13832 =   20^3 +   18^3 =   24^3 +    2^3
   4:     20683 =   24^3 +   19^3 =   27^3 +   10^3
   5:     32832 =   30^3 +   18^3 =   32^3 +    4^3
   6:     39312 =   33^3 +   15^3 =   34^3 +    2^3
   7:     40033 =   33^3 +   16^3 =   34^3 +    9^3
   8:     46683 =   30^3 +   27^3 =   36^3 +    3^3
   9:     64232 =   36^3 +   26^3 =   39^3 +   17^3
  10:     65728 =   33^3 +   31^3 =   40^3 +   12^3
  11:    110656 =   40^3 +   36^3 =   48^3 +    4^3
  12:    110808 =   45^3 +   27^3 =   48^3 +    6^3
  13:    134379 =   43^3 +   38^3 =   51^3 +   12^3
  14:    149389 =   50^3 +   29^3 =   53^3 +    8^3
  15:    165464 =   48^3 +   38^3 =   54^3 +   20^3
  16:    171288 =   54^3 +   24^3 =   55^3 +   17^3
  17:    195841 =   57^3 +   22^3 =   58^3 +    9^3
  18:    216027 =   59^3 +   22^3 =   60^3 +    3^3
  19:    216125 =   50^3 +   45^3 =   60^3 +    5^3
  20:    262656 =   60^3 +   36^3 =   64^3 +    8^3
  21:    314496 =   66^3 +   30^3 =   68^3 +    4^3
  22:    320264 =   66^3 +   32^3 =   68^3 +   18^3
  23:    327763 =   58^3 +   51^3 =   67^3 +   30^3
  24:    373464 =   60^3 +   54^3 =   72^3 +    6^3
  25:    402597 =   61^3 +   56^3 =   69^3 +   42^3
2000:1671816384 =  944^3 +  940^3 = 1168^3 +  428^3
2001:1672470592 = 1124^3 +  632^3 = 1187^3 +   29^3
2002:1673170856 = 1034^3 +  828^3 = 1164^3 +  458^3
2003:1675045225 = 1081^3 +  744^3 = 1153^3 +  522^3
2004:1675958167 = 1096^3 +  711^3 = 1159^3 +  492^3
2005:1676926719 = 1095^3 +  714^3 = 1188^3 +   63^3
2006:1677646971 =  990^3 +  891^3 = 1188^3 +   99^3

D

High Level Version

Translation of: Python
void main() /*@safe*/ {
    import std.stdio, std.range, std.algorithm, std.typecons, std.string;

    auto iCubes = iota(1u, 1201u).map!(x => tuple(x, x ^^ 3));
    bool[Tuple!(uint, uint)][uint] sum2cubes;
    foreach (i, immutable i3; iCubes)
        foreach (j, immutable j3; iCubes[i .. $])
            sum2cubes[i3 + j3][tuple(i, j)] = true;

    const taxis = sum2cubes.byKeyValue.filter!(p => p.value.length > 1)
                  .array.schwartzSort!(p => p.key).release;

    foreach (/*immutable*/ const r; [[0, 25], [2000 - 1, 2000 + 6]]) {
        foreach (immutable i, const t; taxis[r[0] .. r[1]])
            writefln("%4d: %10d =%-(%s =%)", i + r[0] + 1, t.key,
                     t.value.keys.sort().map!q{"%4d^3 + %4d^3".format(a[])});
        writeln;
    }
}
Output:
   1:       1729 =   1^3 +   12^3 =   9^3 +   10^3
   2:       4104 =   2^3 +   16^3 =   9^3 +   15^3
   3:      13832 =   2^3 +   24^3 =  18^3 +   20^3
   4:      20683 =  10^3 +   27^3 =  19^3 +   24^3
   5:      32832 =   4^3 +   32^3 =  18^3 +   30^3
   6:      39312 =   2^3 +   34^3 =  15^3 +   33^3
   7:      40033 =   9^3 +   34^3 =  16^3 +   33^3
   8:      46683 =   3^3 +   36^3 =  27^3 +   30^3
   9:      64232 =  17^3 +   39^3 =  26^3 +   36^3
  10:      65728 =  12^3 +   40^3 =  31^3 +   33^3
  11:     110656 =   4^3 +   48^3 =  36^3 +   40^3
  12:     110808 =   6^3 +   48^3 =  27^3 +   45^3
  13:     134379 =  12^3 +   51^3 =  38^3 +   43^3
  14:     149389 =   8^3 +   53^3 =  29^3 +   50^3
  15:     165464 =  20^3 +   54^3 =  38^3 +   48^3
  16:     171288 =  17^3 +   55^3 =  24^3 +   54^3
  17:     195841 =   9^3 +   58^3 =  22^3 +   57^3
  18:     216027 =   3^3 +   60^3 =  22^3 +   59^3
  19:     216125 =   5^3 +   60^3 =  45^3 +   50^3
  20:     262656 =   8^3 +   64^3 =  36^3 +   60^3
  21:     314496 =   4^3 +   68^3 =  30^3 +   66^3
  22:     320264 =  18^3 +   68^3 =  32^3 +   66^3
  23:     327763 =  30^3 +   67^3 =  51^3 +   58^3
  24:     373464 =   6^3 +   72^3 =  54^3 +   60^3
  25:     402597 =  42^3 +   69^3 =  56^3 +   61^3

2000: 1671816384 = 428^3 + 1168^3 = 940^3 +  944^3
2001: 1672470592 =  29^3 + 1187^3 = 632^3 + 1124^3
2002: 1673170856 = 458^3 + 1164^3 = 828^3 + 1034^3
2003: 1675045225 = 522^3 + 1153^3 = 744^3 + 1081^3
2004: 1675958167 = 492^3 + 1159^3 = 711^3 + 1096^3
2005: 1676926719 =  63^3 + 1188^3 = 714^3 + 1095^3
2006: 1677646971 =  99^3 + 1188^3 = 891^3 +  990^3

Run-time: about 2.9 seconds with dmd compiler.

Heap-Based Version

Translation of: Java
import std.stdio, std.string, std.container;

struct CubeSum {
    ulong x, y, value;

    this(in ulong x_, in ulong y_) pure nothrow @safe @nogc {
        this.x = x_;
        this.y = y_;
        this.value = x_ ^^ 3 + y_ ^^ 3;
    }
}

final class Taxi {
    BinaryHeap!(Array!CubeSum, "a.value > b.value") pq;
    CubeSum last;
    ulong n = 0;

    this() {
        last = nextSum();
    }

    CubeSum nextSum() {
        while (pq.empty || pq.front.value >= n ^^ 3)
            pq.insert(CubeSum(++n, 1));

        auto s = pq.front;
        pq.removeFront;
        if (s.x > s.y + 1)
            pq.insert(CubeSum(s.x, s.y + 1));

        return s;
    }

    CubeSum[] nextTaxi() {
        CubeSum s;
        typeof(return) train;

        while ((s = nextSum).value != last.value)
            last = s;

        train ~= last;

        do {
            train ~= s;
        } while ((s = nextSum).value == last.value);
        last = s;

        return train;
    }
}

void main() {
    auto taxi = new Taxi;

    foreach (immutable i; 1 .. 2007) {
        const t = taxi.nextTaxi;
        if (i > 25 && i < 2000)
            continue;

        writef("%4d: %10d", i, t[0].value);
        foreach (const s; t)
            writef(" = %4d^3 + %4d^3", s.x, s.y);
        writeln;
    }
}
Output:
   1:       1729 =   10^3 +    9^3 =   12^3 +    1^3
   2:       4104 =   15^3 +    9^3 =   16^3 +    2^3
   3:      13832 =   20^3 +   18^3 =   24^3 +    2^3
   4:      20683 =   24^3 +   19^3 =   27^3 +   10^3
   5:      32832 =   30^3 +   18^3 =   32^3 +    4^3
   6:      39312 =   33^3 +   15^3 =   34^3 +    2^3
   7:      40033 =   33^3 +   16^3 =   34^3 +    9^3
   8:      46683 =   30^3 +   27^3 =   36^3 +    3^3
   9:      64232 =   39^3 +   17^3 =   36^3 +   26^3
  10:      65728 =   40^3 +   12^3 =   33^3 +   31^3
  11:     110656 =   40^3 +   36^3 =   48^3 +    4^3
  12:     110808 =   45^3 +   27^3 =   48^3 +    6^3
  13:     134379 =   51^3 +   12^3 =   43^3 +   38^3
  14:     149389 =   50^3 +   29^3 =   53^3 +    8^3
  15:     165464 =   48^3 +   38^3 =   54^3 +   20^3
  16:     171288 =   54^3 +   24^3 =   55^3 +   17^3
  17:     195841 =   57^3 +   22^3 =   58^3 +    9^3
  18:     216027 =   59^3 +   22^3 =   60^3 +    3^3
  19:     216125 =   50^3 +   45^3 =   60^3 +    5^3
  20:     262656 =   60^3 +   36^3 =   64^3 +    8^3
  21:     314496 =   66^3 +   30^3 =   68^3 +    4^3
  22:     320264 =   68^3 +   18^3 =   66^3 +   32^3
  23:     327763 =   67^3 +   30^3 =   58^3 +   51^3
  24:     373464 =   60^3 +   54^3 =   72^3 +    6^3
  25:     402597 =   69^3 +   42^3 =   61^3 +   56^3
2000: 1671816384 = 1168^3 +  428^3 =  944^3 +  940^3
2001: 1672470592 = 1124^3 +  632^3 = 1187^3 +   29^3
2002: 1673170856 = 1164^3 +  458^3 = 1034^3 +  828^3
2003: 1675045225 = 1153^3 +  522^3 = 1081^3 +  744^3
2004: 1675958167 = 1159^3 +  492^3 = 1096^3 +  711^3
2005: 1676926719 = 1095^3 +  714^3 = 1188^3 +   63^3
2006: 1677646971 =  990^3 +  891^3 = 1188^3 +   99^3

Run-time: about 0.31 seconds with ldc2 compiler. It's faster than the Java solution.

Low Level Heap-Based Version

Translation of: C
struct Taxicabs {
    alias CubesSumT = uint; // Or ulong.

    static struct Sum {
        CubesSumT value;
        uint x, y;
    }

    // The cubes can be pre-computed if CubesSumT is a BigInt.
    private uint nCubes;
    private Sum[] pq;
    private uint pq_len;

    private void addCube() pure nothrow @safe {
        nCubes = nCubes ? nCubes + 1 : 2;
        if (nCubes < 2)
            return; // 0 or 1 is useless.

        pq_len++;
        if (pq_len >= pq.length)
            pq.length = (pq.length == 0) ? 2 : (pq.length * 2);

        immutable tmp = Sum(CubesSumT(nCubes - 2) ^^ 3 + 1,
                            nCubes - 2, 1);

        // Upheap.
        uint i = pq_len;
        for (; i >= 1 && pq[i >> 1].value > tmp.value; i >>= 1)
            pq[i] = pq[i >> 1];

        pq[i] = tmp;
    }


    private void nextSum() pure nothrow @safe {
        while (!pq_len || pq[1].value >= (nCubes - 1) ^^ 3)
            addCube();

        Sum tmp = pq[0] = pq[1]; //pq[0] always stores last seen value.
        tmp.y++;
        if (tmp.y >= tmp.x) { // Done with this x; throw it away.
            tmp = pq[pq_len];
            pq_len--;
            if (!pq_len)
                return nextSum(); // Refill empty heap.
        } else
            tmp.value += tmp.y ^^ 3 - (tmp.y - 1) ^^ 3;

        // Downheap.
        uint i = 1;
        while (true) {
            uint j = i << 1;
            if (j > pq_len)
                break;
            if (j < pq_len && pq[j + 1].value < pq[j].value)
                j++;
            if (pq[j].value >= tmp.value)
                break;
            pq[i] = pq[j];
            i = j;
        }

        pq[i] = tmp;
    }


    Sum[] nextTaxi(size_t N)(ref Sum[N] hist)
    pure nothrow @safe {
        do {
            nextSum();
        } while (pq[0].value != pq[1].value);

        uint len = 1;
        hist[0] = pq[0];
        do {
            hist[len] = pq[1];
            len++;
            nextSum();
        } while (pq[0].value == pq[1].value);

        return hist[0 .. len];
    }
}


void main() nothrow {
    import core.stdc.stdio;

    Taxicabs t;
    Taxicabs.Sum[3] x;

    foreach (immutable uint i; 1 .. 2007) {
        const triples = t.nextTaxi(x);
        if (i > 25 && i < 2000)
            continue;
        printf("%4u: %10lu", i, triples[0].value);
        foreach_reverse (const s; triples)
            printf(" = %4u^3 + %4u^3", s.x, s.y);
        '\n'.putchar;
    }
}
Output:
   1:       1729 =   12^3 +    1^3 =   10^3 +    9^3
   2:       4104 =   15^3 +    9^3 =   16^3 +    2^3
   3:      13832 =   20^3 +   18^3 =   24^3 +    2^3
   4:      20683 =   27^3 +   10^3 =   24^3 +   19^3
   5:      32832 =   30^3 +   18^3 =   32^3 +    4^3
   6:      39312 =   33^3 +   15^3 =   34^3 +    2^3
   7:      40033 =   33^3 +   16^3 =   34^3 +    9^3
   8:      46683 =   30^3 +   27^3 =   36^3 +    3^3
   9:      64232 =   36^3 +   26^3 =   39^3 +   17^3
  10:      65728 =   33^3 +   31^3 =   40^3 +   12^3
  11:     110656 =   40^3 +   36^3 =   48^3 +    4^3
  12:     110808 =   45^3 +   27^3 =   48^3 +    6^3
  13:     134379 =   43^3 +   38^3 =   51^3 +   12^3
  14:     149389 =   50^3 +   29^3 =   53^3 +    8^3
  15:     165464 =   48^3 +   38^3 =   54^3 +   20^3
  16:     171288 =   54^3 +   24^3 =   55^3 +   17^3
  17:     195841 =   57^3 +   22^3 =   58^3 +    9^3
  18:     216027 =   59^3 +   22^3 =   60^3 +    3^3
  19:     216125 =   50^3 +   45^3 =   60^3 +    5^3
  20:     262656 =   60^3 +   36^3 =   64^3 +    8^3
  21:     314496 =   66^3 +   30^3 =   68^3 +    4^3
  22:     320264 =   66^3 +   32^3 =   68^3 +   18^3
  23:     327763 =   58^3 +   51^3 =   67^3 +   30^3
  24:     373464 =   60^3 +   54^3 =   72^3 +    6^3
  25:     402597 =   61^3 +   56^3 =   69^3 +   42^3
2000: 1671816384 = 1168^3 +  428^3 =  944^3 +  940^3
2001: 1672470592 = 1124^3 +  632^3 = 1187^3 +   29^3
2002: 1673170856 = 1034^3 +  828^3 = 1164^3 +  458^3
2003: 1675045225 = 1153^3 +  522^3 = 1081^3 +  744^3
2004: 1675958167 = 1096^3 +  711^3 = 1159^3 +  492^3
2005: 1676926719 = 1188^3 +   63^3 = 1095^3 +  714^3
2006: 1677646971 =  990^3 +  891^3 = 1188^3 +   99^3

Run-time: about 0.08 seconds with ldc2 compiler.

DCL

We invoke external utility SORT which I suppose technically speaking is not a formal part of the language but is darn handy at times;

$ close /nolog sums_of_cubes
$ on control_y then $ goto clean
$ open /write sums_of_cubes sums_of_cubes.txt
$ i = 1
$ loop1:
$  write sys$output i
$  j = 1
$  loop2:
$   sum = i * i * i + j * j * j
$   if sum .lt. 0
$   then
$    write sys$output "overflow at ", j
$    goto next_i
$   endif
$   write sums_of_cubes f$fao( "!10SL,!10SL,!10SL", sum, i, j )
$   j = j + 1
$   if j .le. i then $ goto loop2
$ next_i:
$  i = i + 1
$  if i .le. 1289 then $ goto loop1  ! cube_root of 2^31-1
$ close sums_of_cubes
$ sort sums_of_cubes.txt sorted_sums_of_cubes.txt
$ close /nolog sorted_sums_of_cubes
$ open sorted_sums_of_cubes sorted_sums_of_cubes.txt
$ count = 0
$ read sorted_sums_of_cubes prev_prev_line  ! need to detect when there are more than just 2 different sums, e.g. 456
$ prev_prev_sum = f$element( 0, ",", f$edit( prev_prev_line, "collapse" ))
$ read sorted_sums_of_cubes prev_line
$ prev_sum = f$element( 0,",", f$edit( prev_line, "collapse" ))
$ loop3:
$  read /end_of_file = done sorted_sums_of_cubes line
$  sum = f$element( 0, ",", f$edit( line, "collapse" ))
$  if sum .eqs. prev_sum
$  then
$   if sum .nes. prev_prev_sum then $ count = count + 1
$   int_sum = f$integer( sum )
$   i1 = f$integer( f$element( 1, ",", prev_line ))
$   j1 = f$integer( f$element( 2, ",", prev_line ))
$   i2 = f$integer( f$element( 1, ",", line ))
$   j2 = f$integer( f$element( 2, ",", line ))
$   if count .le. 25 .or. ( count .ge. 2000 .and. count .le. 2006 ) then -
$    write sys$output f$fao( "!4SL:!11SL =!5SL^3 +!5SL^3 =!5SL^3 +!5SL^3", count, int_sum, i1, j1, i2, j2 )
$  endif
$  prev_prev_line = prev_line
$  prev_prev_sum = prev_sum
$  prev_line = line
$  prev_sum = sum
$  goto loop3
$ done:
$ close sorted_sums_of_cubes
$ exit
$
$ clean:
$ close /nolog sorted_sums_of_cubes
$ close /nolog sums_of_cubes
Output:
$ @taxicab_numbers
   1:       1729 =   10^3 +    9^3 =   12^3 +    1^3
   2:       4104 =   15^3 +    9^3 =   16^3 +    2^3
   3:      13832 =   20^3 +   18^3 =   24^3 +    2^3
   4:      20683 =   24^3 +   19^3 =   27^3 +   10^3
   5:      32832 =   30^3 +   18^3 =   32^3 +    4^3
   6:      39312 =   33^3 +   15^3 =   34^3 +    2^3
   7:      40033 =   33^3 +   16^3 =   34^3 +    9^3
   8:      46683 =   30^3 +   27^3 =   36^3 +    3^3
   9:      64232 =   36^3 +   26^3 =   39^3 +   17^3
  10:      65728 =   33^3 +   31^3 =   40^3 +   12^3
  11:     110656 =   40^3 +   36^3 =   48^3 +    4^3
  12:     110808 =   45^3 +   27^3 =   48^3 +    6^3
  13:     134379 =   43^3 +   38^3 =   51^3 +   12^3
  14:     149389 =   50^3 +   29^3 =   53^3 +    8^3
  15:     165464 =   48^3 +   38^3 =   54^3 +   20^3
  16:     171288 =   54^3 +   24^3 =   55^3 +   17^3
  17:     195841 =   57^3 +   22^3 =   58^3 +    9^3
  18:     216027 =   59^3 +   22^3 =   60^3 +    3^3
  19:     216125 =   50^3 +   45^3 =   60^3 +    5^3
  20:     262656 =   60^3 +   36^3 =   64^3 +    8^3
  21:     314496 =   66^3 +   30^3 =   68^3 +    4^3
  22:     320264 =   66^3 +   32^3 =   68^3 +   18^3
  23:     327763 =   58^3 +   51^3 =   67^3 +   30^3
  24:     373464 =   60^3 +   54^3 =   72^3 +    6^3
  25:     402597 =   61^3 +   56^3 =   69^3 +   42^3
2000: 1671816384 =  944^3 +  940^3 = 1168^3 +  428^3
2001: 1672470592 = 1124^3 +  632^3 = 1187^3 +   29^3
2002: 1673170856 = 1034^3 +  828^3 = 1164^3 +  458^3
2003: 1675045225 = 1081^3 +  744^3 = 1153^3 +  522^3
2004: 1675958167 = 1096^3 +  711^3 = 1159^3 +  492^3
2005: 1676926719 = 1095^3 +  714^3 = 1188^3 +   63^3
2006: 1677646971 =  990^3 +  891^3 = 1188^3 +   99^3


Delphi

See Pascal.


EasyLang

fastfunc taxi n m .
   repeat
      m += 1
      m3 = m * m * m
      until m3 >= n / 2
      p = m
      repeat
         p += 1
         h = m3 + p * p * p
         if h = n
            return m
         .
         until h >= n
      .
   .
   return 0
.
func part2 n m .
   return floor (0.5 + pow (n - m * m * m) (1 / 3))
.
repeat
   n += 1
   t1 = taxi n 0
   if t1 > 0
      t2 = taxi n t1
      if t2 > 0
         cnt += 1
         write n & " = "
         write t1 & "³ + " & part2 n t1 & "³ = "
         print t2 & "³ + " & part2 n t2 & "³"
      .
   .
   until cnt = 25
.
Output:
1729 = 1³ + 12³ = 9³ + 10³
4104 = 2³ + 16³ = 9³ + 15³
13832 = 2³ + 24³ = 18³ + 20³
20683 = 10³ + 27³ = 19³ + 24³
32832 = 4³ + 32³ = 18³ + 30³
39312 = 2³ + 34³ = 15³ + 33³
40033 = 9³ + 34³ = 16³ + 33³
46683 = 3³ + 36³ = 27³ + 30³
64232 = 17³ + 39³ = 26³ + 36³
65728 = 12³ + 40³ = 31³ + 33³
110656 = 4³ + 48³ = 36³ + 40³
110808 = 6³ + 48³ = 27³ + 45³
134379 = 12³ + 51³ = 38³ + 43³
149389 = 8³ + 53³ = 29³ + 50³
165464 = 20³ + 54³ = 38³ + 48³
171288 = 17³ + 55³ = 24³ + 54³
195841 = 9³ + 58³ = 22³ + 57³
216027 = 3³ + 60³ = 22³ + 59³
216125 = 5³ + 60³ = 45³ + 50³
262656 = 8³ + 64³ = 36³ + 60³
314496 = 4³ + 68³ = 30³ + 66³
320264 = 18³ + 68³ = 32³ + 66³
327763 = 30³ + 67³ = 51³ + 58³
373464 = 6³ + 72³ = 54³ + 60³
402597 = 42³ + 69³ = 56³ + 61³

EchoLisp

Using the heap library, and a heap to store the taxicab numbers. For taxi tuples - decomposition in more than two sums - we use the group function which transforms a list ( 3 5 5 6 8 ...) into ((3) (5 5) (6) ...).

(require '(heap compile))

(define (scube a b) (+ (* a a a) (* b b b)))
(compile 'scube "-f") ; "-f" means : no bigint, no rational used

;; is n - a^3 a cube  b^3?
;; if yes return b, else #f
(define (taxi? n a (b 0))
	(set! b (cbrt (- n (* a a a)))) ;; cbrt is ∛
	(when (and (< b a) (integer? b)) b))
(compile 'taxi? "-f")

#|-------------------
looking for taxis
--------------------|#
;; remove from heap until heap-top >= a
;; when twins are removed, it is a taxicab number : push it
;; at any time (top stack) = last removed

(define (clean-taxi H limit: a  min-of-heap: htop)
		(when (and htop (> a htop))
				(when (!= (stack-top S) htop) (pop S))
				(push S htop)
				(heap-pop H)
				(clean-taxi H  a (heap-top H))))
(compile 'clean-taxi "-f")
				
;; loop on a and b, b <=a , until n taxicabs found
(define (taxicab (n 2100))
	(for ((a (in-naturals)))
		(clean-taxi H (* a a a) (heap-top H))
		#:break (> (stack-length S) n)
		(for ((b a))
			(heap-push H (scube a b)))))
				
#|------------------
printing taxis
---------------------|#
;; string of all decompositions
(define (taxi->string i n)
	(string-append (format "%d. %d " (1+ i) n)
	(for/string ((a (cbrt n)))
		#:when (taxi? n a)
		(format " = %4d^3 + %4d^3" a (taxi? n a)))))
		
(define (taxi-print taxis (nfrom 0) (nto 26))
		(for ((i (in-naturals nfrom)) (taxi (sublist taxis nfrom nto)))
		(writeln (taxi->string i (first taxi)))))
Output:
(define S (stack 'S)) ;; to push taxis
(define H (make-heap < )) ;; make min heap of all scubes

(taxicab 2100)
(define taxis (group (stack->list S)))
(taxi-print taxis )

1. 1729 =   10^3 +    9^3 =   12^3 +    1^3    
2. 4104 =   15^3 +    9^3 =   16^3 +    2^3    
3. 13832 =   20^3 +   18^3 =   24^3 +    2^3    
4. 20683 =   24^3 +   19^3 =   27^3 +   10^3    
#| ... |#  
24. 373464 =   60^3 +   54^3 =   72^3 +    6^3    
25. 402597 =   61^3 +   56^3 =   69^3 +   42^3    
26. 439101 =   69^3 +   48^3 =   76^3 +    5^3    

(taxi-print taxis 1999 2006)
2000. 1671816384 = 944^3 + 940^3 = 1168^3 + 428^3    
2001. 1672470592 = 1124^3 + 632^3 = 1187^3 +   29^3    
2002. 1673170856 = 1034^3 + 828^3 = 1164^3 + 458^3    
2003. 1675045225 = 1081^3 + 744^3 = 1153^3 + 522^3    
2004. 1675958167 = 1096^3 + 711^3 = 1159^3 + 492^3    
2005. 1676926719 = 1095^3 + 714^3 = 1188^3 +   63^3    
2006. 1677646971 = 990^3 + 891^3 = 1188^3 +   99^3    

;; extra bonus : print all taxis which are triplets
(define (taxi-tuples taxis (nfrom 0) (nto 2000))
		(for ((i (in-naturals nfrom)) (taxi (sublist taxis nfrom nto)))
		#:when (> (length taxi) 1) ;; filter for tuples is here
		(writeln (taxi->string i (first taxi)))))
		
(taxi-tuples taxis)

455. 87539319 = 414^3 + 255^3 = 423^3 + 228^3 = 436^3 + 167^3    
535. 119824488 = 428^3 + 346^3 = 492^3 +   90^3 = 493^3 +   11^3    
588. 143604279 = 423^3 + 408^3 = 460^3 + 359^3 = 522^3 + 111^3    
655. 175959000 = 525^3 + 315^3 = 552^3 + 198^3 = 560^3 +   70^3    
888. 327763000 = 580^3 + 510^3 = 661^3 + 339^3 = 670^3 + 300^3    
1299. 700314552 = 828^3 + 510^3 = 846^3 + 456^3 = 872^3 + 334^3    
1398. 804360375 = 920^3 + 295^3 = 927^3 + 198^3 = 930^3 +   15^3    
1515. 958595904 = 856^3 + 692^3 = 984^3 + 180^3 = 986^3 +   22^3    
1660. 1148834232 = 846^3 + 816^3 = 920^3 + 718^3 = 1044^3 + 222^3    
1837. 1407672000 = 1050^3 + 630^3 = 1104^3 + 396^3 = 1120^3 + 140^3

Elixir

defmodule Taxicab do
  def numbers(n \\ 1200) do
    (for i <- 1..n, j <- i..n, do: {i,j})
    |> Enum.group_by(fn {i,j} -> i*i*i + j*j*j end)
    |> Enum.filter(fn {_,v} -> length(v)>1 end)
    |> Enum.sort
  end
end

nums = Taxicab.numbers |> Enum.with_index
Enum.each(nums, fn {x,i} ->
  if i in 0..24 or i in 1999..2005 do
    IO.puts "#{i+1} : #{inspect x}"
  end
end)
Output:
1 : {1729, [{9, 10}, {1, 12}]}
2 : {4104, [{9, 15}, {2, 16}]}
3 : {13832, [{18, 20}, {2, 24}]}
4 : {20683, [{19, 24}, {10, 27}]}
5 : {32832, [{18, 30}, {4, 32}]}
6 : {39312, [{15, 33}, {2, 34}]}
7 : {40033, [{16, 33}, {9, 34}]}
8 : {46683, [{27, 30}, {3, 36}]}
9 : {64232, [{26, 36}, {17, 39}]}
10 : {65728, [{31, 33}, {12, 40}]}
11 : {110656, [{36, 40}, {4, 48}]}
12 : {110808, [{27, 45}, {6, 48}]}
13 : {134379, [{38, 43}, {12, 51}]}
14 : {149389, [{29, 50}, {8, 53}]}
15 : {165464, [{38, 48}, {20, 54}]}
16 : {171288, [{24, 54}, {17, 55}]}
17 : {195841, [{22, 57}, {9, 58}]}
18 : {216027, [{22, 59}, {3, 60}]}
19 : {216125, [{45, 50}, {5, 60}]}
20 : {262656, [{36, 60}, {8, 64}]}
21 : {314496, [{30, 66}, {4, 68}]}
22 : {320264, [{32, 66}, {18, 68}]}
23 : {327763, [{51, 58}, {30, 67}]}
24 : {373464, [{54, 60}, {6, 72}]}
25 : {402597, [{56, 61}, {42, 69}]}
2000 : {1671816384, [{940, 944}, {428, 1168}]}
2001 : {1672470592, [{632, 1124}, {29, 1187}]}
2002 : {1673170856, [{828, 1034}, {458, 1164}]}
2003 : {1675045225, [{744, 1081}, {522, 1153}]}
2004 : {1675958167, [{711, 1096}, {492, 1159}]}
2005 : {1676926719, [{714, 1095}, {63, 1188}]}
2006 : {1677646971, [{891, 990}, {99, 1188}]}


Forth

Works with: gforth version 0.7.3
variable taxicablist
variable searched-cubessum
73 constant max-constituent \ uses magic numbers

: cube dup dup * * ;
: cubessum cube swap cube + ;

: ?taxicab ( a b -- c d true | false )
\ does exist an (c, d) such that c^3+d^3 = a^3+b^3 ?
  2dup cubessum searched-cubessum !
  dup 1- rot 1+ do   \ c is possibly in [a+1 b-2]
    dup i 1+ do      \ d is possibly in [c+1 b-1]
      j i cubessum searched-cubessum @ = if drop j i true unloop unloop exit then
    loop
  loop drop false ;

: swap-taxi ( n -- )
  dup 5 cells + swap do
    i @     i 5 cells - @     i !     i 5 cells - !
  cell +loop ;

: bubble-taxicablist
  here 5 cells - taxicablist @ = if exit then        \ not for the first one
  taxicablist @ here 5 cells - do
    i @     i 5 cells - @     > if unloop exit then  \ no (more) need to reorder
    i swap-taxi
  5 cells -loop ;

: store-taxicab ( a b c d -- )
  2dup cubessum , swap , , swap , ,
  bubble-taxicablist ;

: build-taxicablist
  here taxicablist !
  max-constituent 3 - 1 do         \ a in [ 1 ; max-3 ]
    i 3 + max-constituent swap do  \ b in [ a+3 ; max ]
      j i ?taxicab if j i store-taxicab then
    loop
  loop ;

: .nextcube cell + dup @ . ." ^3 " ;
: .taxi
  dup @ .
  ." = " .nextcube ." + " .nextcube ." = " .nextcube ." + " .nextcube
  drop ;

: taxicab 5 cells * taxicablist @ + ;

: list-taxicabs ( n -- )
  0 do
    cr I 1+ . ." : "
    I taxicab .taxi
  loop ;

build-taxicablist
25 list-taxicabs
Output:
1 : 1729 = 1 ^3 + 12 ^3 = 9 ^3 + 10 ^3 
2 : 4104 = 2 ^3 + 16 ^3 = 9 ^3 + 15 ^3 
3 : 13832 = 2 ^3 + 24 ^3 = 18 ^3 + 20 ^3 
4 : 20683 = 10 ^3 + 27 ^3 = 19 ^3 + 24 ^3 
5 : 32832 = 4 ^3 + 32 ^3 = 18 ^3 + 30 ^3 
6 : 39312 = 2 ^3 + 34 ^3 = 15 ^3 + 33 ^3 
7 : 40033 = 9 ^3 + 34 ^3 = 16 ^3 + 33 ^3 
8 : 46683 = 3 ^3 + 36 ^3 = 27 ^3 + 30 ^3 
9 : 64232 = 17 ^3 + 39 ^3 = 26 ^3 + 36 ^3 
10 : 65728 = 12 ^3 + 40 ^3 = 31 ^3 + 33 ^3 
11 : 110656 = 4 ^3 + 48 ^3 = 36 ^3 + 40 ^3 
12 : 110808 = 6 ^3 + 48 ^3 = 27 ^3 + 45 ^3 
13 : 134379 = 12 ^3 + 51 ^3 = 38 ^3 + 43 ^3 
14 : 149389 = 8 ^3 + 53 ^3 = 29 ^3 + 50 ^3 
15 : 165464 = 20 ^3 + 54 ^3 = 38 ^3 + 48 ^3 
16 : 171288 = 17 ^3 + 55 ^3 = 24 ^3 + 54 ^3 
17 : 195841 = 9 ^3 + 58 ^3 = 22 ^3 + 57 ^3 
18 : 216027 = 3 ^3 + 60 ^3 = 22 ^3 + 59 ^3 
19 : 216125 = 5 ^3 + 60 ^3 = 45 ^3 + 50 ^3 
20 : 262656 = 8 ^3 + 64 ^3 = 36 ^3 + 60 ^3 
21 : 314496 = 4 ^3 + 68 ^3 = 30 ^3 + 66 ^3 
22 : 320264 = 18 ^3 + 68 ^3 = 32 ^3 + 66 ^3 
23 : 327763 = 30 ^3 + 67 ^3 = 51 ^3 + 58 ^3 
24 : 373464 = 6 ^3 + 72 ^3 = 54 ^3 + 60 ^3 
25 : 402597 = 42 ^3 + 69 ^3 = 56 ^3 + 61 ^3  ok

One can use 1200 as magic number rather than 73 and display 2006 taxicab numbers ('2006 list-taxicabs') but the 10 triple taxicabs will slide the count...


Fortran

! A non-bruteforce approach
     PROGRAM POOKA
      IMPLICIT NONE
!
! PARAMETER definitions
!
      INTEGER , PARAMETER  ::  NVARS = 25
!
! Local variables
!
      REAL  ::  f1
      REAL  ::  f2
      INTEGER  ::  hits
      INTEGER  ::  s
      INTEGER  ::  TAXICAB

      hits = 0
      s = 0
      f1 = SECOND()
      DO WHILE ( hits<NVARS )
         s = s + 1
         hits = hits + TAXICAB(s)
      END DO
      f2 = SECOND()
      PRINT * , 'elapsed time = ' , f2 - f1 , 'For ' , NVARS , ' Variables'
      STOP
      END PROGRAM POOKA
 
      FUNCTION TAXICAB(N)
      IMPLICIT NONE
!
! Dummy arguments
!
      INTEGER  ::  N
      INTEGER  ::  TAXICAB
      INTENT (IN) N
!
! Local variables
!
      INTEGER  ::  holder
      INTEGER  ::  oldx
      INTEGER  ::  oldy
      INTEGER  ::  s
      INTEGER  ::  x
      INTEGER  ::  y
      real*8,parameter :: xpon=(1.0D0/3.0D0)
!
      x = 0
      holder = 0
      oldx = 0
      oldy = 0
      TAXICAB = 0
      y = INT(N**xpon)
      DO WHILE ( x<=y )
         s = x**3 + y**3
         IF( s<N )THEN
            x = x + 1
         ELSE IF( s>N )THEN
            y = y - 1
         ELSE
            IF( holder==s )THEN ! Print the last value and this one that correspond
               WRITE(6 , 34)s , '(' , x**3 , y**3 , ')' , '(' , oldx**3 , oldy**3 , ')'
 34            FORMAT(1x , i12 , 10x , 1A1 , i12 , 2x , i12 , 1A1 , 10x , 1A1 , i12 , 2x ,&
                    & i12 , 1A1)
               TAXICAB = 1  ! Indicate that we found a Taxi Number
            END IF
            holder = s      ! Set to the number that appears a potential cab number
            oldx = x       ! Retain the values for the 2 cubes
            oldy = y
            x = x + 1       ! Keep looking
            y = y - 1
         END IF
      END DO
      RETURN
      END FUNCTION TAXICAB
Output:
  Print first 25 numbers
         1729          (         729          1000)          (           1          1728)
         4104          (         729          3375)          (           8          4096)
        13832          (        5832          8000)          (           8         13824)
        20683          (        6859         13824)          (        1000         19683)
        32832          (        5832         27000)          (          64         32768)
        39312          (        3375         35937)          (           8         39304)
        40033          (        4096         35937)          (         729         39304)
        46683          (       19683         27000)          (          27         46656)
        64232          (       17576         46656)          (        4913         59319)
        65728          (       29791         35937)          (        1728         64000)
       110656          (       46656         64000)          (          64        110592)
       110808          (       19683         91125)          (         216        110592)
       134379          (       54872         79507)          (        1728        132651)
       149389          (       24389        125000)          (         512        148877)
       165464          (       54872        110592)          (        8000        157464)
       171288          (       13824        157464)          (        4913        166375)
       195841          (       10648        185193)          (         729        195112)
       216027          (       10648        205379)          (          27        216000)
       216125          (       91125        125000)          (         125        216000)
       262656          (       46656        216000)          (         512        262144)
       314496          (       27000        287496)          (          64        314432)
       320264          (       32768        287496)          (        5832        314432)
       327763          (      132651        195112)          (       27000        300763)
       373464          (      157464        216000)          (         216        373248)
       402597          (      175616        226981)          (       74088        328509)
 elapsed time =    4.68750000E-02 For           25  Variables

      -----   2000 to 2006 numbers
   1671816384          (   830584000     841232384)          (    78402752    1593413632)               
   1672470592          (   252435968    1420034624)          (       24389    1672446203)              
   1673170856          (   567663552    1105507304)          (    96071912    1577098944)             
   1675045225          (   411830784    1263214441)          (   142236648    1532808577)               
   1675958167          (   359425431    1316532736)          (   119095488    1556862679)               
   1676926719          (   363994344    1312932375)          (      250047    1676676672)            
   1677646971          (   707347971     970299000)          (      970299    1676676672)

FreeBASIC

' version 11-10-2016
' compile with: fbc -s console

' Brute force

' adopted from "Sorting algorithms/Shell" sort Task
Sub shellsort(s() As String)
  ' sort from lower bound to the highter bound
  Dim As UInteger lb = LBound(s)
  Dim As UInteger ub = UBound(s)
  Dim As Integer done, i, inc = ub - lb

  Do
    inc = inc / 2.2
    If inc < 1 Then inc = 1

    Do
      done = 0
      For i = lb To ub - inc
        If s(i) > s(i + inc) Then
          Swap s(i), s(i + inc)
          done = 1
        End If
      Next
    Loop Until done = 0

  Loop Until inc = 1

End Sub

' ------=< MAIN >=------

Dim As UInteger x, y, count, c, sum
Dim As UInteger cube(1290)
Dim As String result(), str1, str2, str3
Dim As String buf11 = Space(11), buf5 = Space(5)
ReDim result(900000)    ' ~1291*1291\2

' set up the cubes
Print : Print " Calculate cubes"
For x = 1 To 1290
  cube(x) = x*x*x
Next

' combine and store
Print : Print " Combine cubes"
For x = 1 To 1290
  For y = x To 1290
    sum = cube(x)+cube(y)
    RSet buf11, Str(sum) : str1 = buf11
    RSet buf5, Str(x) : str2 = buf5
    RSet buf5, Str(y) : Str3 = buf5
    result(count)=buf11 + " = " + str2 + " ^ 3 + " + str3 + " ^ 3"
    count = count +1
  Next
Next

count= count -1
ReDim Preserve result(count) ' trim the array

Print : Print " Sort (takes some time)"
shellsort(result())   ' sort

Print : Print " Find the Taxicab numbers"
c = 1 ' start at index 1
For x = 0 To count -1
  ' find sums that match
  If Left(result(x), 11) = Left(result(x + 1), 11) Then
    result(c) = result(x)
    y = x +1
    Do    ' merge the other solution(s)
      result(c) = result(c) + Mid(result(y), 12)
      y = y +1
    Loop Until Left(result(x), 11) <> Left(result(y), 11)
    x = y -1 ' let x point to last match result
    c = c +1 
  End If
Next

c = c -1
Print : Print " "; c; " Taxicab numbers found"
ReDim Preserve result(c) ' trim the array again

cls
Print : Print " Print first 25 numbers" : Print
For x = 1 To 25
  Print result(x)
Next

Print : Print " The 2000th to the 2006th" : Print
For x = 2000 To 2006
  Print result(x)
Next


' empty keyboard buffer
While Inkey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End
Output:
  Print first 25 numbers

       1729 =     1 ^ 3 +    12 ^ 3 =     9 ^ 3 +    10 ^ 3
       4104 =     2 ^ 3 +    16 ^ 3 =     9 ^ 3 +    15 ^ 3
      13832 =     2 ^ 3 +    24 ^ 3 =    18 ^ 3 +    20 ^ 3
      20683 =    10 ^ 3 +    27 ^ 3 =    19 ^ 3 +    24 ^ 3
      32832 =     4 ^ 3 +    32 ^ 3 =    18 ^ 3 +    30 ^ 3
      39312 =     2 ^ 3 +    34 ^ 3 =    15 ^ 3 +    33 ^ 3
      40033 =     9 ^ 3 +    34 ^ 3 =    16 ^ 3 +    33 ^ 3
      46683 =     3 ^ 3 +    36 ^ 3 =    27 ^ 3 +    30 ^ 3
      64232 =    17 ^ 3 +    39 ^ 3 =    26 ^ 3 +    36 ^ 3
      65728 =    12 ^ 3 +    40 ^ 3 =    31 ^ 3 +    33 ^ 3
     110656 =     4 ^ 3 +    48 ^ 3 =    36 ^ 3 +    40 ^ 3
     110808 =     6 ^ 3 +    48 ^ 3 =    27 ^ 3 +    45 ^ 3
     134379 =    12 ^ 3 +    51 ^ 3 =    38 ^ 3 +    43 ^ 3
     149389 =     8 ^ 3 +    53 ^ 3 =    29 ^ 3 +    50 ^ 3
     165464 =    20 ^ 3 +    54 ^ 3 =    38 ^ 3 +    48 ^ 3
     171288 =    17 ^ 3 +    55 ^ 3 =    24 ^ 3 +    54 ^ 3
     195841 =     9 ^ 3 +    58 ^ 3 =    22 ^ 3 +    57 ^ 3
     216027 =     3 ^ 3 +    60 ^ 3 =    22 ^ 3 +    59 ^ 3
     216125 =     5 ^ 3 +    60 ^ 3 =    45 ^ 3 +    50 ^ 3
     262656 =     8 ^ 3 +    64 ^ 3 =    36 ^ 3 +    60 ^ 3
     314496 =     4 ^ 3 +    68 ^ 3 =    30 ^ 3 +    66 ^ 3
     320264 =    18 ^ 3 +    68 ^ 3 =    32 ^ 3 +    66 ^ 3
     327763 =    30 ^ 3 +    67 ^ 3 =    51 ^ 3 +    58 ^ 3
     373464 =     6 ^ 3 +    72 ^ 3 =    54 ^ 3 +    60 ^ 3
     402597 =    42 ^ 3 +    69 ^ 3 =    56 ^ 3 +    61 ^ 3

 The 2000th to the 2006th

 1671816384 =   428 ^ 3 +  1168 ^ 3 =   940 ^ 3 +   944 ^ 3
 1672470592 =    29 ^ 3 +  1187 ^ 3 =   632 ^ 3 +  1124 ^ 3
 1673170856 =   458 ^ 3 +  1164 ^ 3 =   828 ^ 3 +  1034 ^ 3
 1675045225 =   522 ^ 3 +  1153 ^ 3 =   744 ^ 3 +  1081 ^ 3
 1675958167 =   492 ^ 3 +  1159 ^ 3 =   711 ^ 3 +  1096 ^ 3
 1676926719 =    63 ^ 3 +  1188 ^ 3 =   714 ^ 3 +  1095 ^ 3
 1677646971 =    99 ^ 3 +  1188 ^ 3 =   891 ^ 3 +   990 ^ 3

Go

package main

import (
	"container/heap"
	"fmt"
	"strings"
)

type CubeSum struct {
	x, y  uint16
	value uint64
}

func (c *CubeSum) fixvalue() { c.value = cubes[c.x] + cubes[c.y] }

type CubeSumHeap []*CubeSum

func (h CubeSumHeap) Len() int            { return len(h) }
func (h CubeSumHeap) Less(i, j int) bool  { return h[i].value < h[j].value }
func (h CubeSumHeap) Swap(i, j int)       { h[i], h[j] = h[j], h[i] }
func (h *CubeSumHeap) Push(x interface{}) { (*h) = append(*h, x.(*CubeSum)) }
func (h *CubeSumHeap) Pop() interface{} {
	x := (*h)[len(*h)-1]
	*h = (*h)[:len(*h)-1]
	return x
}

type TaxicabGen struct {
	n int
	h CubeSumHeap
}

var cubes []uint64 // cubes[i] == i*i*i
func cubesExtend(i int) {
	for n := uint64(len(cubes)); n <= uint64(i); n++ {
		cubes = append(cubes, n*n*n)
	}
}

func (g *TaxicabGen) min() CubeSum {
	for len(g.h) == 0 || g.h[0].value > cubes[g.n] {
		g.n++
		cubesExtend(g.n)
		heap.Push(&g.h, &CubeSum{uint16(g.n), 1, cubes[g.n] + 1})
	}
	// Note, we use g.h[0] to "peek" at the min heap entry.
	c := *(g.h[0])
	if c.y+1 <= c.x {
		// Instead of Pop and Push we modify in place and fix.
		g.h[0].y++
		g.h[0].fixvalue()
		heap.Fix(&g.h, 0)
	} else {
		heap.Pop(&g.h)
	}
	return c
}

// Originally this was just: type Taxicab [2]CubeSum
// and we always returned two sums. Now we return all the sums.
type Taxicab []CubeSum

func (t Taxicab) String() string {
	var b strings.Builder
	fmt.Fprintf(&b, "%12d", t[0].value)
	for _, p := range t {
		fmt.Fprintf(&b, " =%5d³ +%5d³", p.x, p.y)
	}
	return b.String()
}

func (g *TaxicabGen) Next() Taxicab {
	a, b := g.min(), g.min()
	for a.value != b.value {
		a, b = b, g.min()
	}
	//return Taxicab{a,b}

	// Originally this just returned Taxicab{a,b} and we didn't look
	// further into the heap. Since we start by looking at the next
	// pair, that is okay until the first Taxicab number with four
	// ways of expressing the cube, which doesn't happen until the
	// 97,235th Taxicab:
	//     6963472309248 = 16630³ + 13322³ = 18072³ + 10200³
	//                   = 18948³ +  5436³ = 19083³ +  2421³
	// Now we return all ways so we need to peek into the heap.
	t := Taxicab{a, b}
	for g.h[0].value == b.value {
		t = append(t, g.min())
	}
	return t
}

func main() {
	const (
		low  = 25
		mid  = 2e3
		high = 4e4
	)
	var tg TaxicabGen
	firstn := 3 // To show the first triple, quadruple, etc
	for i := 1; i <= high+6; i++ {
		t := tg.Next()
		switch {
		case len(t) >= firstn:
			firstn++
			fallthrough
		case i <= low || (mid <= i && i <= mid+6) || i >= high:
			//fmt.Printf("h:%-4d  ", len(tg.h))
			fmt.Printf("%5d: %v\n", i, t)
		}
	}
}
Output:
    1:         1729 =   12³ +    1³ =   10³ +    9³
    2:         4104 =   16³ +    2³ =   15³ +    9³
    3:        13832 =   24³ +    2³ =   20³ +   18³
    4:        20683 =   27³ +   10³ =   24³ +   19³
    5:        32832 =   32³ +    4³ =   30³ +   18³
    6:        39312 =   34³ +    2³ =   33³ +   15³
    7:        40033 =   34³ +    9³ =   33³ +   16³
    8:        46683 =   36³ +    3³ =   30³ +   27³
    9:        64232 =   36³ +   26³ =   39³ +   17³
   10:        65728 =   40³ +   12³ =   33³ +   31³
   11:       110656 =   48³ +    4³ =   40³ +   36³
   12:       110808 =   48³ +    6³ =   45³ +   27³
   13:       134379 =   51³ +   12³ =   43³ +   38³
   14:       149389 =   53³ +    8³ =   50³ +   29³
   15:       165464 =   54³ +   20³ =   48³ +   38³
   16:       171288 =   55³ +   17³ =   54³ +   24³
   17:       195841 =   58³ +    9³ =   57³ +   22³
   18:       216027 =   60³ +    3³ =   59³ +   22³
   19:       216125 =   60³ +    5³ =   50³ +   45³
   20:       262656 =   64³ +    8³ =   60³ +   36³
   21:       314496 =   68³ +    4³ =   66³ +   30³
   22:       320264 =   66³ +   32³ =   68³ +   18³
   23:       327763 =   58³ +   51³ =   67³ +   30³
   24:       373464 =   72³ +    6³ =   60³ +   54³
   25:       402597 =   69³ +   42³ =   61³ +   56³
  455:     87539319 =  436³ +  167³ =  423³ +  228³ =  414³ +  255³
 2000:   1671816384 = 1168³ +  428³ =  944³ +  940³
 2001:   1672470592 = 1187³ +   29³ = 1124³ +  632³
 2002:   1673170856 = 1164³ +  458³ = 1034³ +  828³
 2003:   1675045225 = 1081³ +  744³ = 1153³ +  522³
 2004:   1675958167 = 1096³ +  711³ = 1159³ +  492³
 2005:   1676926719 = 1188³ +   63³ = 1095³ +  714³
 2006:   1677646971 =  990³ +  891³ = 1188³ +   99³
40000: 976889700163 = 8659³ + 6894³ = 9891³ + 2098³
40001: 976942087381 = 7890³ + 7861³ = 8680³ + 6861³
40002: 976946344920 = 9476³ + 5014³ = 9798³ + 3312³
40003: 976962998375 = 9912³ + 1463³ = 8415³ + 7250³
40004: 976974757064 = 9365³ + 5379³ = 9131³ + 5997³
40005: 977025552984 = 9894³ + 2040³ = 9792³ + 3366³
40006: 977104161000 = 9465³ + 5055³ = 9920³ +  970³

Haskell

import Data.List (groupBy, sortOn, tails, transpose)
import Data.Function (on)

--------------------- TAXICAB NUMBERS --------------------

taxis :: Int -> [[(Int, ((Int, Int), (Int, Int)))]]
taxis nCubes =
  filter ((> 1) . length) $
  groupBy (on (==) fst) $
  sortOn fst
    [ (fst x + fst y, (x, y))
    | (x:t) <- tails $ ((^ 3) >>= (,)) <$> [1 .. nCubes] 
    , y <- t ]

--------------------------- TEST -------------------------
main :: IO ()
main =
  mapM_ putStrLn $
  concat <$>
  transpose
    (((<$>) =<< flip justifyRight ' ' . maximum . (length <$>)) <$>
     transpose (taxiRow <$> (take 25 xs <> take 7 (drop 1999 xs))))
  where
    xs = zip [1 ..] (taxis 1200)
    justifyRight n c = (drop . length) <*> (replicate n c <>)

------------------------- DISPLAY ------------------------
taxiRow :: (Int, [(Int, ((Int, Int), (Int, Int)))]) -> [String]
taxiRow (n, [(a, ((axc, axr), (ayc, ayr))), (b, ((bxc, bxr), (byc, byr)))]) =
  concat
    [ [show n, ". ", show a, " = "]
    , term axr axc " + "
    , term ayr ayc "  or  "
    , term bxr bxc " + "
    , term byr byc []
    ]
  where
    term r c l = ["(", show r, "^3=", show c, ")", l]
Output:
   1.       1729 = (  1^3=        1) + (  12^3=      1728)  or  (  9^3=      729) + (  10^3=      1000)
   2.       4104 = (  2^3=        8) + (  16^3=      4096)  or  (  9^3=      729) + (  15^3=      3375)
   3.      13832 = (  2^3=        8) + (  24^3=     13824)  or  ( 18^3=     5832) + (  20^3=      8000)
   4.      20683 = ( 10^3=     1000) + (  27^3=     19683)  or  ( 19^3=     6859) + (  24^3=     13824)
   5.      32832 = (  4^3=       64) + (  32^3=     32768)  or  ( 18^3=     5832) + (  30^3=     27000)
   6.      39312 = (  2^3=        8) + (  34^3=     39304)  or  ( 15^3=     3375) + (  33^3=     35937)
   7.      40033 = (  9^3=      729) + (  34^3=     39304)  or  ( 16^3=     4096) + (  33^3=     35937)
   8.      46683 = (  3^3=       27) + (  36^3=     46656)  or  ( 27^3=    19683) + (  30^3=     27000)
   9.      64232 = ( 17^3=     4913) + (  39^3=     59319)  or  ( 26^3=    17576) + (  36^3=     46656)
  10.      65728 = ( 12^3=     1728) + (  40^3=     64000)  or  ( 31^3=    29791) + (  33^3=     35937)
  11.     110656 = (  4^3=       64) + (  48^3=    110592)  or  ( 36^3=    46656) + (  40^3=     64000)
  12.     110808 = (  6^3=      216) + (  48^3=    110592)  or  ( 27^3=    19683) + (  45^3=     91125)
  13.     134379 = ( 12^3=     1728) + (  51^3=    132651)  or  ( 38^3=    54872) + (  43^3=     79507)
  14.     149389 = (  8^3=      512) + (  53^3=    148877)  or  ( 29^3=    24389) + (  50^3=    125000)
  15.     165464 = ( 20^3=     8000) + (  54^3=    157464)  or  ( 38^3=    54872) + (  48^3=    110592)
  16.     171288 = ( 17^3=     4913) + (  55^3=    166375)  or  ( 24^3=    13824) + (  54^3=    157464)
  17.     195841 = (  9^3=      729) + (  58^3=    195112)  or  ( 22^3=    10648) + (  57^3=    185193)
  18.     216027 = (  3^3=       27) + (  60^3=    216000)  or  ( 22^3=    10648) + (  59^3=    205379)
  19.     216125 = (  5^3=      125) + (  60^3=    216000)  or  ( 45^3=    91125) + (  50^3=    125000)
  20.     262656 = (  8^3=      512) + (  64^3=    262144)  or  ( 36^3=    46656) + (  60^3=    216000)
  21.     314496 = (  4^3=       64) + (  68^3=    314432)  or  ( 30^3=    27000) + (  66^3=    287496)
  22.     320264 = ( 18^3=     5832) + (  68^3=    314432)  or  ( 32^3=    32768) + (  66^3=    287496)
  23.     327763 = ( 30^3=    27000) + (  67^3=    300763)  or  ( 51^3=   132651) + (  58^3=    195112)
  24.     373464 = (  6^3=      216) + (  72^3=    373248)  or  ( 54^3=   157464) + (  60^3=    216000)
  25.     402597 = ( 42^3=    74088) + (  69^3=    328509)  or  ( 56^3=   175616) + (  61^3=    226981)
2000. 1671816384 = (428^3= 78402752) + (1168^3=1593413632)  or  (940^3=830584000) + ( 944^3= 841232384)
2001. 1672470592 = ( 29^3=    24389) + (1187^3=1672446203)  or  (632^3=252435968) + (1124^3=1420034624)
2002. 1673170856 = (458^3= 96071912) + (1164^3=1577098944)  or  (828^3=567663552) + (1034^3=1105507304)
2003. 1675045225 = (522^3=142236648) + (1153^3=1532808577)  or  (744^3=411830784) + (1081^3=1263214441)
2004. 1675958167 = (492^3=119095488) + (1159^3=1556862679)  or  (711^3=359425431) + (1096^3=1316532736)
2005. 1676926719 = ( 63^3=   250047) + (1188^3=1676676672)  or  (714^3=363994344) + (1095^3=1312932375)
2006. 1677646971 = ( 99^3=   970299) + (1188^3=1676676672)  or  (891^3=707347971) + ( 990^3= 970299000)

J

cubes=: 3^~1+i.100 NB. first 100 cubes
triples=: /:~ ~. ,/ (+ , /:~@,)"0/~cubes NB. ordered pairs of cubes (each with their sum)
candidates=: ;({."#. <@(0&#`({.@{.(;,)<@}."1)@.(1<#))/. ])triples

NB. we just want the first 25 taxicab numbers
25{.(,.~ <@>:@i.@#) candidates
┌──┬──────┬────────────┬─────────────┐
1 1729  1 1728      729 1000     
├──┼──────┼────────────┼─────────────┤
2 4104  8 4096      729 3375     
├──┼──────┼────────────┼─────────────┤
3 13832 8 13824     5832 8000    
├──┼──────┼────────────┼─────────────┤
4 20683 1000 19683  6859 13824   
├──┼──────┼────────────┼─────────────┤
5 32832 64 32768    5832 27000   
├──┼──────┼────────────┼─────────────┤
6 39312 8 39304     3375 35937   
├──┼──────┼────────────┼─────────────┤
7 40033 729 39304   4096 35937   
├──┼──────┼────────────┼─────────────┤
8 46683 27 46656    19683 27000  
├──┼──────┼────────────┼─────────────┤
9 64232 4913 59319  17576 46656  
├──┼──────┼────────────┼─────────────┤
1065728 1728 64000  29791 35937  
├──┼──────┼────────────┼─────────────┤
1111065664 110592   46656 64000  
├──┼──────┼────────────┼─────────────┤
12110808216 110592  19683 91125  
├──┼──────┼────────────┼─────────────┤
131343791728 132651 54872 79507  
├──┼──────┼────────────┼─────────────┤
14149389512 148877  24389 125000 
├──┼──────┼────────────┼─────────────┤
151654648000 157464 54872 110592 
├──┼──────┼────────────┼─────────────┤
161712884913 166375 13824 157464 
├──┼──────┼────────────┼─────────────┤
17195841729 195112  10648 185193 
├──┼──────┼────────────┼─────────────┤
1821602727 216000   10648 205379 
├──┼──────┼────────────┼─────────────┤
19216125125 216000  91125 125000 
├──┼──────┼────────────┼─────────────┤
20262656512 262144  46656 216000 
├──┼──────┼────────────┼─────────────┤
2131449664 314432   27000 287496 
├──┼──────┼────────────┼─────────────┤
223202645832 314432 32768 287496 
├──┼──────┼────────────┼─────────────┤
2332776327000 300763132651 195112
├──┼──────┼────────────┼─────────────┤
24373464216 373248  157464 216000
├──┼──────┼────────────┼─────────────┤
2540259774088 328509175616 226981
└──┴──────┴────────────┴─────────────┘

Explanation:

First, generate 100 cubes.

Then, form a 3 column table of unique rows: sum, small cube, large cube

Then, gather rows where the first entry is the same. Keep the ones with at least two such entries (sorted by ascending order of sum).

Then, place an counting index (starting from 1) in front of each row, so the columns are now: counting index, sum, small cube, large cube.

Note that the cube root of the 25th entry is slightly smaller than 74, so testing against the first 100 cubes is more than sufficient.

Note that here we have elected to show the constituent cubes as themselves rather than as expressions involving their cube roots.

Extra credit:

   x:each 7 {. 1999 }. (,.~ <@>:@i.@#) ;({."#. <@(0&#`({.@{.(;,)<@}."1)@.(1<#))/. ])/:~~.,/(+,/:~@,)"0/~3^~1+i.10000
┌────┬──────────┬────────────────────┬────────────────────┬┐
2000167181638478402752 1593413632 830584000 841232384 ││
├────┼──────────┼────────────────────┼────────────────────┼┤
2001167247059224389 1672446203    252435968 1420034624││
├────┼──────────┼────────────────────┼────────────────────┼┤
2002167317085696071912 1577098944 567663552 1105507304││
├────┼──────────┼────────────────────┼────────────────────┼┤
20031675045225142236648 1532808577411830784 1263214441││
├────┼──────────┼────────────────────┼────────────────────┼┤
20041675958167119095488 1556862679359425431 1316532736││
├────┼──────────┼────────────────────┼────────────────────┼┤
20051676926719250047 1676676672   363994344 1312932375││
├────┼──────────┼────────────────────┼────────────────────┼┤
20061677646971970299 1676676672   707347971 970299000 ││
└────┴──────────┴────────────────────┴────────────────────┴┘

The extra blank box at the end is because when tackling this large of a data set, some sums can be achieved by three different pairs of cubes.

Java

import java.util.PriorityQueue;
import java.util.ArrayList;
import java.util.List;
import java.util.Iterator;

class CubeSum implements Comparable<CubeSum> {
	public long x, y, value;

	public CubeSum(long x, long y) {
		this.x = x;
		this.y = y;
		this.value = x*x*x + y*y*y;
	}

	public String toString() {
		return String.format("%4d^3 + %4d^3", x, y);
	}

	public int compareTo(CubeSum that) {
		return value < that.value ? -1 : value > that.value ? 1 : 0;
	}
}

class SumIterator implements Iterator<CubeSum> {
	PriorityQueue<CubeSum> pq = new PriorityQueue<CubeSum>();
	long n = 0;

	public boolean hasNext() { return true; }
	public CubeSum next() {
		while (pq.size() == 0 || pq.peek().value >= n*n*n)
			pq.add(new CubeSum(++n, 1));

		CubeSum s = pq.remove();
		if (s.x > s.y + 1) pq.add(new CubeSum(s.x, s.y+1));

		return s;
	}
}

class TaxiIterator implements Iterator<List<CubeSum>> {
	Iterator<CubeSum> sumIterator = new SumIterator();
	CubeSum last = sumIterator.next();

	public boolean hasNext() { return true; }
	public List<CubeSum> next() {
		CubeSum s;
		List<CubeSum> train = new ArrayList<CubeSum>();

		while ((s = sumIterator.next()).value != last.value)
			last = s;

		train.add(last);

		do { train.add(s); } while ((s = sumIterator.next()).value == last.value);
		last = s;

		return train;
	}
}
	
public class Taxi {
	public static final void main(String[] args) {
		Iterator<List<CubeSum>> taxi = new TaxiIterator();

		for (int i = 1; i <= 2006; i++) {
			List<CubeSum> t = taxi.next();
			if (i > 25 && i < 2000) continue;

			System.out.printf("%4d: %10d", i, t.get(0).value);
			for (CubeSum s: t)
				System.out.print(" = " + s);
			System.out.println();
		}
	}
}
Output:
   1:       1729 =   10^3 +    9^3 =   12^3 +    1^3
   2:       4104 =   15^3 +    9^3 =   16^3 +    2^3
   3:      13832 =   20^3 +   18^3 =   24^3 +    2^3
   4:      20683 =   24^3 +   19^3 =   27^3 +   10^3
   5:      32832 =   30^3 +   18^3 =   32^3 +    4^3
   6:      39312 =   33^3 +   15^3 =   34^3 +    2^3
   7:      40033 =   34^3 +    9^3 =   33^3 +   16^3
   8:      46683 =   30^3 +   27^3 =   36^3 +    3^3
   9:      64232 =   36^3 +   26^3 =   39^3 +   17^3
  10:      65728 =   33^3 +   31^3 =   40^3 +   12^3
  11:     110656 =   40^3 +   36^3 =   48^3 +    4^3
  12:     110808 =   45^3 +   27^3 =   48^3 +    6^3
  13:     134379 =   43^3 +   38^3 =   51^3 +   12^3
  14:     149389 =   50^3 +   29^3 =   53^3 +    8^3
  15:     165464 =   48^3 +   38^3 =   54^3 +   20^3
  16:     171288 =   54^3 +   24^3 =   55^3 +   17^3
  17:     195841 =   57^3 +   22^3 =   58^3 +    9^3
  18:     216027 =   59^3 +   22^3 =   60^3 +    3^3
  19:     216125 =   50^3 +   45^3 =   60^3 +    5^3
  20:     262656 =   60^3 +   36^3 =   64^3 +    8^3
  21:     314496 =   66^3 +   30^3 =   68^3 +    4^3
  22:     320264 =   66^3 +   32^3 =   68^3 +   18^3
  23:     327763 =   58^3 +   51^3 =   67^3 +   30^3
  24:     373464 =   60^3 +   54^3 =   72^3 +    6^3
  25:     402597 =   61^3 +   56^3 =   69^3 +   42^3
2000: 1671816384 = 1168^3 +  428^3 =  944^3 +  940^3
2001: 1672470592 = 1124^3 +  632^3 = 1187^3 +   29^3
2002: 1673170856 = 1164^3 +  458^3 = 1034^3 +  828^3
2003: 1675045225 = 1153^3 +  522^3 = 1081^3 +  744^3
2004: 1675958167 = 1159^3 +  492^3 = 1096^3 +  711^3
2005: 1676926719 = 1095^3 +  714^3 = 1188^3 +   63^3
2006: 1677646971 =  990^3 +  891^3 = 1188^3 +   99^3

JavaScript

var n3s = [],
    s3s = {}
for (var n = 1, e = 1200; n < e; n += 1) n3s[n] = n * n * n
for (var a = 1; a < e - 1; a += 1) {
    var a3 = n3s[a]
    for (var b = a; b < e; b += 1) {
        var b3 = n3s[b]
        var s3 = a3 + b3,
            abs = s3s[s3]
        if (!abs) s3s[s3] = abs = []
        abs.push([a, b])
    }
}

var i = 0
for (var s3 in s3s) {
    var abs = s3s[s3]
    if (abs.length < 2) continue
    i += 1
    if (abs.length == 2 && i > 25 && i < 2000) continue
    if (i > 2006) break
    document.write(i, ': ', s3)
    for (var ab of abs) {
        document.write(' = ', ab[0], '<sup>3</sup>+', ab[1], '<sup>3</sup>')
    }
    document.write('<br>')
}
Output:
1: 1729 = 13+123 = 93+103
2: 4104 = 23+163 = 93+153
3: 13832 = 23+243 = 183+203
4: 20683 = 103+273 = 193+243
5: 32832 = 43+323 = 183+303
6: 39312 = 23+343 = 153+333
7: 40033 = 93+343 = 163+333
8: 46683 = 33+363 = 273+303
9: 64232 = 173+393 = 263+363
10: 65728 = 123+403 = 313+333
11: 110656 = 43+483 = 363+403
12: 110808 = 63+483 = 273+453
13: 134379 = 123+513 = 383+433
14: 149389 = 83+533 = 293+503
15: 165464 = 203+543 = 383+483
16: 171288 = 173+553 = 243+543
17: 195841 = 93+583 = 223+573
18: 216027 = 33+603 = 223+593
19: 216125 = 53+603 = 453+503
20: 262656 = 83+643 = 363+603
21: 314496 = 43+683 = 303+663
22: 320264 = 183+683 = 323+663
23: 327763 = 303+673 = 513+583
24: 373464 = 63+723 = 543+603
25: 402597 = 423+693 = 563+613
455: 87539319 = 1673+4363 = 2283+4233 = 2553+4143
535: 119824488 = 113+4933 = 903+4923 = 3463+4283
588: 143604279 = 1113+5223 = 3593+4603 = 4083+4233
655: 175959000 = 703+5603 = 1983+5523 = 3153+5253
888: 327763000 = 3003+6703 = 3393+6613 = 5103+5803
1299: 700314552 = 3343+8723 = 4563+8463 = 5103+8283
1398: 804360375 = 153+9303 = 1983+9273 = 2953+9203
1515: 958595904 = 223+9863 = 1803+9843 = 6923+8563
1660: 1148834232 = 2223+10443 = 7183+9203 = 8163+8463
1837: 1407672000 = 1403+11203 = 3963+11043 = 6303+10503
2000: 1671816384 = 4283+11683 = 9403+9443
2001: 1672470592 = 293+11873 = 6323+11243
2002: 1673170856 = 4583+11643 = 8283+10343
2003: 1675045225 = 5223+11533 = 7443+10813
2004: 1675958167 = 4923+11593 = 7113+10963
2005: 1676926719 = 633+11883 = 7143+10953
2006: 1677646971 = 993+11883 = 8913+9903

jq

Works with: jq version 1.4
# Output: an array of the form [i^3 + j^3, [i, j]] sorted by the sum.
# Only cubes of 1 to ($in-1) are considered; the listing is therefore truncated
# as it might not capture taxicab numbers greater than $in ^ 3.
def sum_of_two_cubes:
  def cubed: .*.*.;
  . as $in
  | (cubed + 1) as $limit
  | [range(1;$in) as $i | range($i;$in) as $j

  | [ ($i|cubed) + ($j|cubed), [$i, $j] ] ] | sort
  | map( select( .[0] < $limit ) );

# Output a stream of triples [t, d1, d2], in order of t,
# where t is a taxicab number, and d1 and d2 are distinct
# decompositions [i,j] with i^3 + j^3 == t.
# The stream includes each taxicab number once only.
#
def taxicabs0:
  sum_of_two_cubes as $sums
  | range(1;$sums|length) as $i
  | if $sums[$i][0] == $sums[$i-1][0]
      and ($i==1 or $sums[$i][0] != $sums[$i-2][0])
    then [$sums[$i][0], $sums[$i-1][1], $sums[$i][1]]
    else empty
    end;

# Output a stream of $n taxicab triples: [t, d1, d2] as described above,
# without repeating t.
def taxicabs:
  # If your jq includes until/2 then the following definition
  # can be omitted:
  def until(cond; next):
    def _until: if cond then . else (next|_until) end;  _until;
  . as $n
  | [10, ($n / 10 | floor)] | max as $increment
  | [20, ($n / 2 | floor)] | max
  | [ ., [taxicabs0] ]
  | until( .[1] | length >= $m; (.[0] + $increment) | [., [taxicabs0]] )
  | .[1][0:$n] ;

The task

2006 | taxicabs as $t
| (range(0;25), range(1999;2006)) as $i
| "\($i+1): \($t[$i][0]) ~ \($t[$i][1]) and \($t[$i][2])"
Output:
$ jq -n -r -f Taxicab_numbers.jq
1: 1729 ~ [1,12] and [9,10]
2: 4104 ~ [2,16] and [9,15]
3: 13832 ~ [2,24] and [18,20]
4: 20683 ~ [10,27] and [19,24]
5: 32832 ~ [4,32] and [18,30]
6: 39312 ~ [2,34] and [15,33]
7: 40033 ~ [9,34] and [16,33]
8: 46683 ~ [3,36] and [27,30]
9: 64232 ~ [17,39] and [26,36]
10: 65728 ~ [12,40] and [31,33]
11: 110656 ~ [4,48] and [36,40]
12: 110808 ~ [6,48] and [27,45]
13: 134379 ~ [12,51] and [38,43]
14: 149389 ~ [8,53] and [29,50]
15: 165464 ~ [20,54] and [38,48]
16: 171288 ~ [17,55] and [24,54]
17: 195841 ~ [9,58] and [22,57]
18: 216027 ~ [3,60] and [22,59]
19: 216125 ~ [5,60] and [45,50]
20: 262656 ~ [8,64] and [36,60]
21: 314496 ~ [4,68] and [30,66]
22: 320264 ~ [18,68] and [32,66]
23: 327763 ~ [30,67] and [51,58]
24: 373464 ~ [6,72] and [54,60]
25: 402597 ~ [42,69] and [56,61]
2000: 1671816384 ~ [428,1168] and [940,944]
2001: 1672470592 ~ [29,1187] and [632,1124]
2002: 1673170856 ~ [458,1164] and [828,1034]
2003: 1675045225 ~ [522,1153] and [744,1081]
2004: 1675958167 ~ [492,1159] and [711,1096]
2005: 1676926719 ~ [63,1188] and [714,1095]
2006: 1677646971 ~ [99,1188] and [891,990]

Julia

Translation of: Python
using Printf, DataStructures, IterTools

function findtaxinumbers(nmax::Integer)
    cube2n = Dict{Int,Int}(x ^ 3 => x for x in 0:nmax)
    sum2cubes = DefaultDict{Int,Set{NTuple{2,Int}}}(Set{NTuple{2,Int}})
    for ((c1, _), (c2, _)) in product(cube2n, cube2n)
        if c1  c2
            push!(sum2cubes[c1 + c2], (cube2n[c1], cube2n[c2]))
        end
    end

    taxied = collect((k, v) for (k, v) in sum2cubes if length(v)  2)
    return sort!(taxied, by = first)
end
taxied = findtaxinumbers(1200)

for (ith, (cube, set)) in zip(1:25, taxied[1:25])
    @printf "%2i: %7i = %s\n" ith cube join(set, ", ")
    # println(ith, ": ", cube, " = ", join(set, ", "))
end
println("...")
for (ith, (cube, set)) in zip(2000:2006, taxied[2000:2006])
    @printf "%-4i: %i = %s\n" ith cube join(set, ", ")
end

# version 2
function findtaxinumbers(nmax::Integer)
    cubes, crev = collect(x ^ 3 for x in 1:nmax), Dict{Int,Int}()
    for (x, x3) in enumerate(cubes)
        crev[x3] = x
    end
    sums = collect(x + y for x in cubes for y in cubes if y < x)
    sort!(sums)

    idx = 0
    for i in 2:(endof(sums) - 1)
        if sums[i-1] != sums[i] && sums[i] == sums[i+1]
            idx += 1
            if 25 < idx < 2000 || idx > 2006 continue end
            n, p = sums[i], NTuple{2,Int}[]
            for x in cubes
                n < 2x && break
                if haskey(crev, n - x)
                    push!(p, (crev[x], crev[n - x]))
                end
            end
            @printf "%4d: %10d" idx n
            for x in p @printf(" = %4d ^ 3 + %4d ^ 3", x...) end
            println()
        end
    end
end

findtaxinumbers(1200)
Output:
 1:    1729 = (12, 1), (10, 9)
 2:    4104 = (16, 2), (15, 9)
 3:   13832 = (24, 2), (20, 18)
 4:   20683 = (27, 10), (24, 19)
 5:   32832 = (32, 4), (30, 18)
 6:   39312 = (33, 15), (34, 2)
 7:   40033 = (34, 9), (33, 16)
 8:   46683 = (30, 27), (36, 3)
 9:   64232 = (36, 26), (39, 17)
10:   65728 = (33, 31), (40, 12)
11:  110656 = (48, 4), (40, 36)
12:  110808 = (48, 6), (45, 27)
13:  134379 = (43, 38), (51, 12)
14:  149389 = (50, 29), (53, 8)
15:  165464 = (54, 20), (48, 38)
16:  171288 = (54, 24), (55, 17)
17:  195841 = (57, 22), (58, 9)
18:  216027 = (59, 22), (60, 3)
19:  216125 = (60, 5), (50, 45)
20:  262656 = (64, 8), (60, 36)
21:  314496 = (66, 30), (68, 4)
22:  320264 = (66, 32), (68, 18)
23:  327763 = (67, 30), (58, 51)
24:  373464 = (60, 54), (72, 6)
25:  402597 = (69, 42), (61, 56)
...
2000: 1671816384 = (944, 940), (1168, 428)
2001: 1672470592 = (1124, 632), (1187, 29)
2002: 1673170856 = (1034, 828), (1164, 458)
2003: 1675045225 = (1081, 744), (1153, 522)
2004: 1675958167 = (1159, 492), (1096, 711)
2005: 1676926719 = (1188, 63), (1095, 714)
2006: 1677646971 = (1188, 99), (990, 891)
   1:       1729 =    1 ^ 3 +   12 ^ 3 =    9 ^ 3 +   10 ^ 3
   2:       4104 =    2 ^ 3 +   16 ^ 3 =    9 ^ 3 +   15 ^ 3
   3:      13832 =    2 ^ 3 +   24 ^ 3 =   18 ^ 3 +   20 ^ 3
   4:      20683 =   10 ^ 3 +   27 ^ 3 =   19 ^ 3 +   24 ^ 3
   5:      32832 =    4 ^ 3 +   32 ^ 3 =   18 ^ 3 +   30 ^ 3
   6:      39312 =    2 ^ 3 +   34 ^ 3 =   15 ^ 3 +   33 ^ 3
   7:      40033 =    9 ^ 3 +   34 ^ 3 =   16 ^ 3 +   33 ^ 3
   8:      46683 =    3 ^ 3 +   36 ^ 3 =   27 ^ 3 +   30 ^ 3
   9:      64232 =   17 ^ 3 +   39 ^ 3 =   26 ^ 3 +   36 ^ 3
  10:      65728 =   12 ^ 3 +   40 ^ 3 =   31 ^ 3 +   33 ^ 3
  11:     110656 =    4 ^ 3 +   48 ^ 3 =   36 ^ 3 +   40 ^ 3
  12:     110808 =    6 ^ 3 +   48 ^ 3 =   27 ^ 3 +   45 ^ 3
  13:     134379 =   12 ^ 3 +   51 ^ 3 =   38 ^ 3 +   43 ^ 3
  14:     149389 =    8 ^ 3 +   53 ^ 3 =   29 ^ 3 +   50 ^ 3
  15:     165464 =   20 ^ 3 +   54 ^ 3 =   38 ^ 3 +   48 ^ 3
  16:     171288 =   17 ^ 3 +   55 ^ 3 =   24 ^ 3 +   54 ^ 3
  17:     195841 =    9 ^ 3 +   58 ^ 3 =   22 ^ 3 +   57 ^ 3
  18:     216027 =    3 ^ 3 +   60 ^ 3 =   22 ^ 3 +   59 ^ 3
  19:     216125 =    5 ^ 3 +   60 ^ 3 =   45 ^ 3 +   50 ^ 3
  20:     262656 =    8 ^ 3 +   64 ^ 3 =   36 ^ 3 +   60 ^ 3
  21:     314496 =    4 ^ 3 +   68 ^ 3 =   30 ^ 3 +   66 ^ 3
  22:     320264 =   18 ^ 3 +   68 ^ 3 =   32 ^ 3 +   66 ^ 3
  23:     327763 =   30 ^ 3 +   67 ^ 3 =   51 ^ 3 +   58 ^ 3
  24:     373464 =    6 ^ 3 +   72 ^ 3 =   54 ^ 3 +   60 ^ 3
  25:     402597 =   42 ^ 3 +   69 ^ 3 =   56 ^ 3 +   61 ^ 3
2000: 1671816384 =  428 ^ 3 + 1168 ^ 3 =  940 ^ 3 +  944 ^ 3
2001: 1672470592 =   29 ^ 3 + 1187 ^ 3 =  632 ^ 3 + 1124 ^ 3
2002: 1673170856 =  458 ^ 3 + 1164 ^ 3 =  828 ^ 3 + 1034 ^ 3
2003: 1675045225 =  522 ^ 3 + 1153 ^ 3 =  744 ^ 3 + 1081 ^ 3
2004: 1675958167 =  492 ^ 3 + 1159 ^ 3 =  711 ^ 3 + 1096 ^ 3
2005: 1676926719 =   63 ^ 3 + 1188 ^ 3 =  714 ^ 3 + 1095 ^ 3
2006: 1677646971 =   99 ^ 3 + 1188 ^ 3 =  891 ^ 3 +  990 ^ 3

Kotlin

Translation of: Java
// version 1.0.6

import java.util.PriorityQueue

class CubeSum(val x: Long, val y: Long) : Comparable<CubeSum> {
    val value: Long = x * x * x + y * y * y

    override fun toString() = String.format("%4d^3 + %3d^3", x, y)
 
    override fun compareTo(other: CubeSum) = value.compareTo(other.value)
}

class SumIterator : Iterator<CubeSum> {
    private val pq = PriorityQueue<CubeSum>()
    private var n = 0L
 
    override fun hasNext() = true

    override fun next(): CubeSum {
        while (pq.size == 0 || pq.peek().value >= n * n * n) 
            pq.add(CubeSum(++n, 1))
        val s: CubeSum = pq.remove()
        if (s.x > s.y + 1) pq.add(CubeSum(s.x, s.y + 1))
        return s
    }
}

class TaxiIterator : Iterator<MutableList<CubeSum>> {
    private val sumIterator = SumIterator()
    private var last: CubeSum = sumIterator.next()  

    override fun hasNext() = true

    override fun next(): MutableList<CubeSum> {
        var s: CubeSum = sumIterator.next()
        val train = mutableListOf<CubeSum>()
        while (s.value != last.value) {
            last = s
            s = sumIterator.next()
        }
        train.add(last)
        do {
            train.add(s)
            s = sumIterator.next()
        }
        while (s.value == last.value)
        last = s
        return train
    }
}

fun main(args: Array<String>) {
    val taxi = TaxiIterator()
    for (i in 1..2006) {
        val t = taxi.next()
        if (i in 26 until 2000) continue
        print(String.format("%4d: %10d", i, t[0].value))  
        for (s in t) print("  = $s")
        println()
    }
}
Output:
   1:       1729  =   10^3 +   9^3  =   12^3 +   1^3
   2:       4104  =   15^3 +   9^3  =   16^3 +   2^3
   3:      13832  =   20^3 +  18^3  =   24^3 +   2^3
   4:      20683  =   24^3 +  19^3  =   27^3 +  10^3
   5:      32832  =   30^3 +  18^3  =   32^3 +   4^3
   6:      39312  =   33^3 +  15^3  =   34^3 +   2^3
   7:      40033  =   33^3 +  16^3  =   34^3 +   9^3
   8:      46683  =   30^3 +  27^3  =   36^3 +   3^3
   9:      64232  =   39^3 +  17^3  =   36^3 +  26^3
  10:      65728  =   40^3 +  12^3  =   33^3 +  31^3
  11:     110656  =   40^3 +  36^3  =   48^3 +   4^3
  12:     110808  =   45^3 +  27^3  =   48^3 +   6^3
  13:     134379  =   51^3 +  12^3  =   43^3 +  38^3
  14:     149389  =   50^3 +  29^3  =   53^3 +   8^3
  15:     165464  =   48^3 +  38^3  =   54^3 +  20^3
  16:     171288  =   54^3 +  24^3  =   55^3 +  17^3
  17:     195841  =   57^3 +  22^3  =   58^3 +   9^3
  18:     216027  =   59^3 +  22^3  =   60^3 +   3^3
  19:     216125  =   50^3 +  45^3  =   60^3 +   5^3
  20:     262656  =   60^3 +  36^3  =   64^3 +   8^3
  21:     314496  =   66^3 +  30^3  =   68^3 +   4^3
  22:     320264  =   68^3 +  18^3  =   66^3 +  32^3
  23:     327763  =   67^3 +  30^3  =   58^3 +  51^3
  24:     373464  =   60^3 +  54^3  =   72^3 +   6^3
  25:     402597  =   69^3 +  42^3  =   61^3 +  56^3
2000: 1671816384  = 1168^3 + 428^3  =  944^3 + 940^3
2001: 1672470592  = 1124^3 + 632^3  = 1187^3 +  29^3
2002: 1673170856  = 1164^3 + 458^3  = 1034^3 + 828^3
2003: 1675045225  = 1153^3 + 522^3  = 1081^3 + 744^3
2004: 1675958167  = 1159^3 + 492^3  = 1096^3 + 711^3
2005: 1676926719  = 1095^3 + 714^3  = 1188^3 +  63^3
2006: 1677646971  =  990^3 + 891^3  = 1188^3 +  99^3

Lua

sums, taxis, limit = {}, {}, 1200
for i = 1, limit do
  for j = 1, i-1 do
    sum = i^3 + j^3
    sums[sum] = sums[sum] or {}
    table.insert(sums[sum], i.."^3 + "..j.."^3")
  end
end
for k,v in pairs(sums) do
  if #v > 1 then table.insert(taxis, { sum=k, num=#v, terms=table.concat(v," = ") }) end
end
table.sort(taxis, function(a,b) return a.sum<b.sum end)
for i=1,2006 do
  if i<=25 or i>=2000 or taxis[i].num==3 then
    print(string.format("%4d%s: %10d = %s", i, taxis[i].num==3 and "*" or " ", taxis[i].sum, taxis[i].terms))
  end
end
print("* n=3")
Output:
   1 :       1729 = 10^3 + 9^3 = 12^3 + 1^3
   2 :       4104 = 15^3 + 9^3 = 16^3 + 2^3
   3 :      13832 = 20^3 + 18^3 = 24^3 + 2^3
   4 :      20683 = 24^3 + 19^3 = 27^3 + 10^3
   5 :      32832 = 30^3 + 18^3 = 32^3 + 4^3
   6 :      39312 = 33^3 + 15^3 = 34^3 + 2^3
   7 :      40033 = 33^3 + 16^3 = 34^3 + 9^3
   8 :      46683 = 30^3 + 27^3 = 36^3 + 3^3
   9 :      64232 = 36^3 + 26^3 = 39^3 + 17^3
  10 :      65728 = 33^3 + 31^3 = 40^3 + 12^3
  11 :     110656 = 40^3 + 36^3 = 48^3 + 4^3
  12 :     110808 = 45^3 + 27^3 = 48^3 + 6^3
  13 :     134379 = 43^3 + 38^3 = 51^3 + 12^3
  14 :     149389 = 50^3 + 29^3 = 53^3 + 8^3
  15 :     165464 = 48^3 + 38^3 = 54^3 + 20^3
  16 :     171288 = 54^3 + 24^3 = 55^3 + 17^3
  17 :     195841 = 57^3 + 22^3 = 58^3 + 9^3
  18 :     216027 = 59^3 + 22^3 = 60^3 + 3^3
  19 :     216125 = 50^3 + 45^3 = 60^3 + 5^3
  20 :     262656 = 60^3 + 36^3 = 64^3 + 8^3
  21 :     314496 = 66^3 + 30^3 = 68^3 + 4^3
  22 :     320264 = 66^3 + 32^3 = 68^3 + 18^3
  23 :     327763 = 58^3 + 51^3 = 67^3 + 30^3
  24 :     373464 = 60^3 + 54^3 = 72^3 + 6^3
  25 :     402597 = 61^3 + 56^3 = 69^3 + 42^3
 455*:   87539319 = 414^3 + 255^3 = 423^3 + 228^3 = 436^3 + 167^3
 535*:  119824488 = 428^3 + 346^3 = 492^3 + 90^3 = 493^3 + 11^3
 588*:  143604279 = 423^3 + 408^3 = 460^3 + 359^3 = 522^3 + 111^3
 655*:  175959000 = 525^3 + 315^3 = 552^3 + 198^3 = 560^3 + 70^3
 888*:  327763000 = 580^3 + 510^3 = 661^3 + 339^3 = 670^3 + 300^3
1299*:  700314552 = 828^3 + 510^3 = 846^3 + 456^3 = 872^3 + 334^3
1398*:  804360375 = 920^3 + 295^3 = 927^3 + 198^3 = 930^3 + 15^3
1515*:  958595904 = 856^3 + 692^3 = 984^3 + 180^3 = 986^3 + 22^3
1660*: 1148834232 = 846^3 + 816^3 = 920^3 + 718^3 = 1044^3 + 222^3
1837*: 1407672000 = 1050^3 + 630^3 = 1104^3 + 396^3 = 1120^3 + 140^3
2000 : 1671816384 = 944^3 + 940^3 = 1168^3 + 428^3
2001 : 1672470592 = 1124^3 + 632^3 = 1187^3 + 29^3
2002 : 1673170856 = 1034^3 + 828^3 = 1164^3 + 458^3
2003 : 1675045225 = 1081^3 + 744^3 = 1153^3 + 522^3
2004 : 1675958167 = 1096^3 + 711^3 = 1159^3 + 492^3
2005 : 1676926719 = 1095^3 + 714^3 = 1188^3 + 63^3
2006 : 1677646971 = 990^3 + 891^3 = 1188^3 + 99^3
* n=3

Mathematica /Wolfram Language

findTaxi[n_] := Sort[Keys[Select[Counts[Flatten[Table[x^3 + y^3, {x, 1, n}, {y, x, n}]]], GreaterThan[1]]]];
Take[findTaxiNumbers[100], 25]
found=findTaxiNumbers[1200][[2000 ;; 2005]]
Map[Reduce[x^3 + y^3 == # && x >= y && x > 0 && y > 0, {x, y}, Integers] &, found]
Output:
{1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, 65728, 110656, 110808, 134379, 149389, 165464, 171288, 195841, 216027, 216125, 262656, 314496, 320264, 327763, 373464, 402597}
{1671816384, 1672470592, 1673170856, 1675045225, 1675958167, 1676926719}
{(x == 944 && y == 940) || (x == 1168 && y == 428), 
(x == 1124 && y == 632) || (x == 1187 && y == 29), 
(x == 1034 && y == 828) || (x == 1164 && y == 458),
(x == 1081 && y == 744) || (x == 1153 && y == 522), 
(x == 1096 && y == 711) || (x == 1159 && y == 492), 
(x == 1095 &&  y == 714) || (x == 1188 && y == 63)}

Nim

Translation of: Python

This is a translation of the Python version which uses a heap.

Python generators have been translated as Nim iterators. We used inline iterators as code expansion is not a problem in this case and performances are better. We formatted the output as in the D version.

Execution time is excellent: about 45 ms on our laptop (I5).

import heapqueue, strformat

type

  CubeSum = tuple[x, y, value: int]

# Comparison function needed for the heap queues.
proc `<`(c1, c2: CubeSum): bool = c1.value < c2.value

template cube(n: int): int = n * n * n


iterator cubesum(): CubeSum =
  var queue: HeapQueue[CubeSum]
  var n = 1
  while true:
    while queue.len == 0 or queue[0].value > cube(n):
      queue.push (n, 1, cube(n) + 1)
      inc n
    var s = queue.pop()
    yield s
    inc s.y
    if s.y < s.x: queue.push (s.x, s.y, cube(s.x) + cube(s.y))


iterator taxis(): seq[CubeSum] =
  var result: seq[CubeSum] = @[(0, 0, 0)]
  for s in cubesum():
    if s.value == result[^1].value:
      result.add s
    else:
      if result.len > 1: yield result
      result.setLen(0)
      result.add s      # These two statements are faster than the single result = @[s].


var n = 0
for t in taxis():
  inc n
  if n > 2006: break
  if n <= 25 or n >= 2000:
    stdout.write &"{n:4}: {t[0].value:10}"
    for s in t:
      stdout.write &" = {s.x:4}^3 + {s.y:4}^3"
    echo()
Output:
   1:       1729 =   10^3 +    9^3 =   12^3 +    1^3
   2:       4104 =   15^3 +    9^3 =   16^3 +    2^3
   3:      13832 =   20^3 +   18^3 =   24^3 +    2^3
   4:      20683 =   24^3 +   19^3 =   27^3 +   10^3
   5:      32832 =   30^3 +   18^3 =   32^3 +    4^3
   6:      39312 =   33^3 +   15^3 =   34^3 +    2^3
   7:      40033 =   34^3 +    9^3 =   33^3 +   16^3
   8:      46683 =   30^3 +   27^3 =   36^3 +    3^3
   9:      64232 =   39^3 +   17^3 =   36^3 +   26^3
  10:      65728 =   33^3 +   31^3 =   40^3 +   12^3
  11:     110656 =   40^3 +   36^3 =   48^3 +    4^3
  12:     110808 =   45^3 +   27^3 =   48^3 +    6^3
  13:     134379 =   43^3 +   38^3 =   51^3 +   12^3
  14:     149389 =   50^3 +   29^3 =   53^3 +    8^3
  15:     165464 =   54^3 +   20^3 =   48^3 +   38^3
  16:     171288 =   54^3 +   24^3 =   55^3 +   17^3
  17:     195841 =   58^3 +    9^3 =   57^3 +   22^3
  18:     216027 =   59^3 +   22^3 =   60^3 +    3^3
  19:     216125 =   50^3 +   45^3 =   60^3 +    5^3
  20:     262656 =   60^3 +   36^3 =   64^3 +    8^3
  21:     314496 =   66^3 +   30^3 =   68^3 +    4^3
  22:     320264 =   68^3 +   18^3 =   66^3 +   32^3
  23:     327763 =   67^3 +   30^3 =   58^3 +   51^3
  24:     373464 =   60^3 +   54^3 =   72^3 +    6^3
  25:     402597 =   61^3 +   56^3 =   69^3 +   42^3
2000: 1671816384 =  944^3 +  940^3 = 1168^3 +  428^3
2001: 1672470592 = 1124^3 +  632^3 = 1187^3 +   29^3
2002: 1673170856 = 1034^3 +  828^3 = 1164^3 +  458^3
2003: 1675045225 = 1153^3 +  522^3 = 1081^3 +  744^3
2004: 1675958167 = 1096^3 +  711^3 = 1159^3 +  492^3
2005: 1676926719 = 1095^3 +  714^3 = 1188^3 +   63^3
2006: 1677646971 = 1188^3 +   99^3 =  990^3 +  891^3

PARI/GP

taxicab(n)=my(t); for(k=sqrtnint((n-1)\2,3)+1, sqrtnint(n,3), if(ispower(n-k^3, 3), if(t, return(1), t=1))); 0;
cubes(n)=my(t); for(k=sqrtnint((n-1)\2,3)+1, sqrtnint(n,3), if(ispower(n-k^3, 3, &t), print(n" =  \t"k"^3\t+ "t"^3")))
select(taxicab, [1..402597])
apply(cubes, %);
Output:
%1 = [1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, 65728, 110656, 110808, 134379, 149389, 165464, 171288, 195841, 216027, 216125, 262656, 314496, 320264, 327763, 373464, 402597]
1729 =          10^3    + 9^3
1729 =          12^3    + 1^3
4104 =          15^3    + 9^3
4104 =          16^3    + 2^3
13832 =         20^3    + 18^3
13832 =         24^3    + 2^3
20683 =         24^3    + 19^3
20683 =         27^3    + 10^3
32832 =         30^3    + 18^3
32832 =         32^3    + 4^3
39312 =         33^3    + 15^3
39312 =         34^3    + 2^3
40033 =         33^3    + 16^3
40033 =         34^3    + 9^3
46683 =         30^3    + 27^3
46683 =         36^3    + 3^3
64232 =         36^3    + 26^3
64232 =         39^3    + 17^3
65728 =         33^3    + 31^3
65728 =         40^3    + 12^3
110656 =        40^3    + 36^3
110656 =        48^3    + 4^3
110808 =        45^3    + 27^3
110808 =        48^3    + 6^3
134379 =        43^3    + 38^3
134379 =        51^3    + 12^3
149389 =        50^3    + 29^3
149389 =        53^3    + 8^3
165464 =        48^3    + 38^3
165464 =        54^3    + 20^3
171288 =        54^3    + 24^3
171288 =        55^3    + 17^3
195841 =        57^3    + 22^3
195841 =        58^3    + 9^3
216027 =        59^3    + 22^3
216027 =        60^3    + 3^3
216125 =        50^3    + 45^3
216125 =        60^3    + 5^3
262656 =        60^3    + 36^3
262656 =        64^3    + 8^3
314496 =        66^3    + 30^3
314496 =        68^3    + 4^3
320264 =        66^3    + 32^3
320264 =        68^3    + 18^3
327763 =        58^3    + 51^3
327763 =        67^3    + 30^3
373464 =        60^3    + 54^3
373464 =        72^3    + 6^3
402597 =        61^3    + 56^3
402597 =        69^3    + 42^3

Pascal

Works with: Free Pascal

Brute force: Create all combinations x³+ y³ | y < x one by on and test if there is a combination v < x and v> w > y with the same cube-sum. Combinations to check = n*(n-1)/2.The mean distance of one Combination m is m/2 from m³+1³ to m³+(m-1)³. searchSameSum checks one half of this distance == m/4.So O(n) ~ n³ /8 checks are needed. searchSameSum takes most of the time (>95% ), sorting is neglectable. [[1]]C-Version is ~6 times faster aka 43 vs 247 ms for max = 1290^3. Here limit set to 1190 to just reach the goal of element 2006 ;-) so 200ms are possible. Its impressive, that over all one check takes ~3.5 cpu-cycles on i4330 3.5Ghz

program taxiCabNo;
uses
  sysutils;
type
  tPot3    = Uint32;
  tPot3Sol = record
               p3Sum : tPot3;
               i1,j1,
               i2,j2 : Word;
             end;
 tpPot3    = ^tPot3;
 tpPot3Sol = ^tPot3Sol;

var
//1290^3 = 2'146'689'000 < 2^31-1
//1190 is the magic number of the task ;-)
  pot3 : array[0..1190{1290}] of tPot3;//
  AllSol : array[0..3000] of tpot3Sol;
  AllSolHigh : NativeInt;

procedure SolOut(const s:tpot3Sol;no: NativeInt);
begin
  with s do
    writeln(no:5,p3Sum:12,' = ',j1:5,'^3 +',i1:5,'^3 =',j2:5,'^3 +',i2:5,'^3');
end;

procedure InsertAllSol;

var
  tmp: tpot3Sol;
  p :tpPot3Sol;
  p3Sum: tPot3;
  i: NativeInt;
Begin

  i := AllSolHigh;
  IF i > 0 then
  Begin
    p := @AllSol[i];
    tmp := p^;
    p3Sum := p^.p3Sum;
    //search the right place for insertion
    repeat
      dec(i);
      dec(p);
      IF (p^.p3Sum <= p3Sum) then
        BREAK;
    until  (i<=0);
    IF p^.p3Sum = p3Sum then
      EXIT;
    //free the right place by moving one place up
    inc(i);
    inc(p);
    IF i<AllSolHigh then
    Begin
      move(p^,AllSol[i+1],SizeOf(AllSol[0])*(AllSolHigh-i));
      p^ := tmp;
    end;
  end;
  inc(AllSolHigh);
end;

function searchSameSum(var sol:tpot3Sol):boolean;
//try to find a new combination for the same sum
//within the limits given by lo and hi
var
  Sum,
  SumLo: tPot3;
  hi,lo: NativeInt;
Begin
  with Sol do
  Begin
    Sum := p3Sum;
    lo:= i1;
    hi:= j1;
  end;

  repeat
    //Move hi down
    dec(hi);
    SumLo := Sum-Pot3[hi];
    //Move lo up an check until new combination found or implicite lo> hi 
    repeat
      inc(lo)
    until (SumLo<=Pot3[lo]);
    //found?
    IF SumLo = Pot3[lo] then
      BREAK;
  until lo>=hi;

  IF lo<hi then
  Begin
    sol.i2:= lo;
    sol.j2:= hi;
    searchSameSum := true;
  end
  else
    searchSameSum := false;
end;

procedure Search;
var
  i,j: LongInt;
Begin
  AllSolHigh := 0;
  For j := 2 to High(pot3)-1 do
  Begin
    For i := 1 to j-1 do
    Begin
      with AllSol[AllSolHigh] do
      Begin
        p3Sum:= pot3[i]+pot3[j];
        i1:= i;
        j1:= j;
      end;
      IF searchSameSum(AllSol[AllSolHigh]) then
      BEGIN
        InsertAllSol;
        IF AllSolHigh>High(AllSol) then EXIT;
      end;
    end;
  end;
end;

var
  i: LongInt;
Begin
  For i := Low(pot3) to High(pot3) do
    pot3[i] := i*i*i;
  AllSolHigh := 0;
  Search;
  For i :=    0 to   24 do SolOut(AllSol[i],i+1);
  For i := 1999 to 2005 do SolOut(AllSol[i],i+1);
  writeln('count of solutions         ',AllSolHigh);
end.
    1        1729 =    12^3 +    1^3 =   10^3 +    9^3
    2        4104 =    16^3 +    2^3 =   15^3 +    9^3
    3       13832 =    24^3 +    2^3 =   20^3 +   18^3
......
   24      373464 =    72^3 +    6^3 =   60^3 +   54^3
   25      402597 =    69^3 +   42^3 =   61^3 +   56^3
 2000  1671816384 =  1168^3 +  428^3 =  944^3 +  940^3
 2001  1672470592 =  1187^3 +   29^3 = 1124^3 +  632^3
...
 2005  1676926719 =  1188^3 +   63^3 = 1095^3 +  714^3
 2006  1677646971 =  1188^3 +   99^3 =  990^3 +  891^3
count of solutions         2050
//checks              196438017
real  0m0.196s

Perl

Uses segmentation so memory use is constrained as high values are searched for. Also has parameter to look for Ta(3) and Ta(4) numbers (which is when segmentation is really needed). By default shows the first 25 numbers; with one argument shows that many; with two arguments shows results in the range.

my($beg, $end) = (@ARGV==0) ? (1,25) : (@ARGV==1) ? (1,shift) : (shift,shift);

my $lim = 1e14;  # Ought to be dynamic as should segment size
my @basis = map { $_*$_*$_ } (1 .. int($lim ** (1.0/3.0) + 1));
my $paira = 2;  # We're looking for Ta(2) and larger

my ($segsize, $low, $high, $i) = (500_000_000, 0, 0, 0);

while ($i < $end) {
  $low = $high+1;
  die "lim too low" if $low > $lim;
  $high = $low + $segsize - 1;
  $high = $lim if $high > $lim;
  foreach my $p (_find_pairs_segment(\@basis, $paira, $low, $high,
                 sub { sprintf("%4d^3 + %4d^3", $_[0], $_[1]) })    ) {
    $i++;
    next if $i < $beg;
    last if $i > $end;
    my $n = shift @$p;
    printf "%4d: %10d  = %s\n", $i, $n, join("  = ", @$p);
  }
}

sub _find_pairs_segment {
  my($p, $len, $start, $end, $formatsub) = @_;
  my $plen = $#$p;

  my %allpairs;
  foreach my $i (0 .. $plen) {
    my $pi = $p->[$i];
    next if ($pi+$p->[$plen]) < $start;
    last if (2*$pi) > $end;
    foreach my $j ($i .. $plen) {
      my $sum = $pi + $p->[$j];
      next if $sum < $start;
      last if $sum > $end;
      push @{ $allpairs{$sum} }, $i, $j;
    }
    # If we wanted to save more memory, we could filter and delete every entry
    # where $n < 2 * $p->[$i+1].  This can cut memory use in half, but is slow.
  }

  my @retlist;
  foreach my $list (grep { scalar @$_ >= $len*2 } values %allpairs) {
    my $n = $p->[$list->[0]] + $p->[$list->[1]];
    my @pairlist;
    while (@$list) {
      push @pairlist, $formatsub->(1 + shift @$list, 1 + shift @$list);
    }
    push @retlist, [$n, @pairlist];
  }
  @retlist = sort { $a->[0] <=> $b->[0] } @retlist;
  return @retlist;
}
Output:
   1:       1729  =    1^3 +   12^3  =    9^3 +   10^3
   2:       4104  =    2^3 +   16^3  =    9^3 +   15^3
   3:      13832  =    2^3 +   24^3  =   18^3 +   20^3
   4:      20683  =   10^3 +   27^3  =   19^3 +   24^3
   5:      32832  =    4^3 +   32^3  =   18^3 +   30^3
   6:      39312  =    2^3 +   34^3  =   15^3 +   33^3
   7:      40033  =    9^3 +   34^3  =   16^3 +   33^3
   8:      46683  =    3^3 +   36^3  =   27^3 +   30^3
   9:      64232  =   17^3 +   39^3  =   26^3 +   36^3
  10:      65728  =   12^3 +   40^3  =   31^3 +   33^3
  11:     110656  =    4^3 +   48^3  =   36^3 +   40^3
  12:     110808  =    6^3 +   48^3  =   27^3 +   45^3
  13:     134379  =   12^3 +   51^3  =   38^3 +   43^3
  14:     149389  =    8^3 +   53^3  =   29^3 +   50^3
  15:     165464  =   20^3 +   54^3  =   38^3 +   48^3
  16:     171288  =   17^3 +   55^3  =   24^3 +   54^3
  17:     195841  =    9^3 +   58^3  =   22^3 +   57^3
  18:     216027  =    3^3 +   60^3  =   22^3 +   59^3
  19:     216125  =    5^3 +   60^3  =   45^3 +   50^3
  20:     262656  =    8^3 +   64^3  =   36^3 +   60^3
  21:     314496  =    4^3 +   68^3  =   30^3 +   66^3
  22:     320264  =   18^3 +   68^3  =   32^3 +   66^3
  23:     327763  =   30^3 +   67^3  =   51^3 +   58^3
  24:     373464  =    6^3 +   72^3  =   54^3 +   60^3
  25:     402597  =   42^3 +   69^3  =   56^3 +   61^3

With arguments 2000 2006:

2000: 1671816384  =  428^3 + 1168^3  =  940^3 +  944^3
2001: 1672470592  =   29^3 + 1187^3  =  632^3 + 1124^3
2002: 1673170856  =  458^3 + 1164^3  =  828^3 + 1034^3
2003: 1675045225  =  522^3 + 1153^3  =  744^3 + 1081^3
2004: 1675958167  =  492^3 + 1159^3  =  711^3 + 1096^3
2005: 1676926719  =   63^3 + 1188^3  =  714^3 + 1095^3
2006: 1677646971  =   99^3 + 1188^3  =  891^3 +  990^3

Phix

-- demo\rosetta\Taxicab_numbers.exw
with javascript_semantics
function cube_sums()
    // create cubes and sorted list of cube sums
    sequence cubes = {}, sums = {}
    for i=1 to 1189 do
        atom cube = i * i * i
        sums &= sq_add(cubes,cube)
        cubes &= cube
    end for
    sums = sort(sums) -- (706,266 in total)
    return {cubes,sums}
end function

sequence {cubes, sums} = cube_sums()

atom nm1 = sums[1],
       n = sums[2]
integer idx = 1
printf(1,"First 25 Taxicab Numbers, 2000..2006th, and all interim triples:\n")
for i=3 to length(sums) do
    atom np1 = sums[i]
    if n=np1 and n!=nm1 then
        if idx<=25 
        or (idx>=2000 and idx<=2006)
        or n=sums[i+1] then
            sequence s = {}
            for j=1 to length(cubes) do
                atom x = cubes[j],
                     y = n-x
                if y<x then exit end if
                integer ydx = binary_search(y,cubes)
                if ydx>0 then
                    s = append(s,sprintf("(%3d^3=%9d) + (%4d^3=%10d)",{j,x,ydx,y}))
                end if
            end for
            printf(1,"%4d %10d = %s\n",{idx,n,join(s," or ")})
        end if
        idx += 1
    end if
    nm1 = n
    n = np1
end for
Output:
First 25 Taxicab Numbers, 2000..2006th, and all interim triples:
   1       1729 = (  1^3=        1) + (  12^3=      1728) or (  9^3=      729) + (  10^3=      1000)
   2       4104 = (  2^3=        8) + (  16^3=      4096) or (  9^3=      729) + (  15^3=      3375)
   3      13832 = (  2^3=        8) + (  24^3=     13824) or ( 18^3=     5832) + (  20^3=      8000)
   4      20683 = ( 10^3=     1000) + (  27^3=     19683) or ( 19^3=     6859) + (  24^3=     13824)
   5      32832 = (  4^3=       64) + (  32^3=     32768) or ( 18^3=     5832) + (  30^3=     27000)
   6      39312 = (  2^3=        8) + (  34^3=     39304) or ( 15^3=     3375) + (  33^3=     35937)
   7      40033 = (  9^3=      729) + (  34^3=     39304) or ( 16^3=     4096) + (  33^3=     35937)
   8      46683 = (  3^3=       27) + (  36^3=     46656) or ( 27^3=    19683) + (  30^3=     27000)
   9      64232 = ( 17^3=     4913) + (  39^3=     59319) or ( 26^3=    17576) + (  36^3=     46656)
  10      65728 = ( 12^3=     1728) + (  40^3=     64000) or ( 31^3=    29791) + (  33^3=     35937)
  11     110656 = (  4^3=       64) + (  48^3=    110592) or ( 36^3=    46656) + (  40^3=     64000)
  12     110808 = (  6^3=      216) + (  48^3=    110592) or ( 27^3=    19683) + (  45^3=     91125)
  13     134379 = ( 12^3=     1728) + (  51^3=    132651) or ( 38^3=    54872) + (  43^3=     79507)
  14     149389 = (  8^3=      512) + (  53^3=    148877) or ( 29^3=    24389) + (  50^3=    125000)
  15     165464 = ( 20^3=     8000) + (  54^3=    157464) or ( 38^3=    54872) + (  48^3=    110592)
  16     171288 = ( 17^3=     4913) + (  55^3=    166375) or ( 24^3=    13824) + (  54^3=    157464)
  17     195841 = (  9^3=      729) + (  58^3=    195112) or ( 22^3=    10648) + (  57^3=    185193)
  18     216027 = (  3^3=       27) + (  60^3=    216000) or ( 22^3=    10648) + (  59^3=    205379)
  19     216125 = (  5^3=      125) + (  60^3=    216000) or ( 45^3=    91125) + (  50^3=    125000)
  20     262656 = (  8^3=      512) + (  64^3=    262144) or ( 36^3=    46656) + (  60^3=    216000)
  21     314496 = (  4^3=       64) + (  68^3=    314432) or ( 30^3=    27000) + (  66^3=    287496)
  22     320264 = ( 18^3=     5832) + (  68^3=    314432) or ( 32^3=    32768) + (  66^3=    287496)
  23     327763 = ( 30^3=    27000) + (  67^3=    300763) or ( 51^3=   132651) + (  58^3=    195112)
  24     373464 = (  6^3=      216) + (  72^3=    373248) or ( 54^3=   157464) + (  60^3=    216000)
  25     402597 = ( 42^3=    74088) + (  69^3=    328509) or ( 56^3=   175616) + (  61^3=    226981)
 455   87539319 = (167^3=  4657463) + ( 436^3=  82881856) or (228^3= 11852352) + ( 423^3=  75686967) or (255^3= 16581375) + ( 414^3=  70957944)
 535  119824488 = ( 11^3=     1331) + ( 493^3= 119823157) or ( 90^3=   729000) + ( 492^3= 119095488) or (346^3= 41421736) + ( 428^3=  78402752)
 588  143604279 = (111^3=  1367631) + ( 522^3= 142236648) or (359^3= 46268279) + ( 460^3=  97336000) or (408^3= 67917312) + ( 423^3=  75686967)
 655  175959000 = ( 70^3=   343000) + ( 560^3= 175616000) or (198^3=  7762392) + ( 552^3= 168196608) or (315^3= 31255875) + ( 525^3= 144703125)
 888  327763000 = (300^3= 27000000) + ( 670^3= 300763000) or (339^3= 38958219) + ( 661^3= 288804781) or (510^3=132651000) + ( 580^3= 195112000)
1299  700314552 = (334^3= 37259704) + ( 872^3= 663054848) or (456^3= 94818816) + ( 846^3= 605495736) or (510^3=132651000) + ( 828^3= 567663552)
1398  804360375 = ( 15^3=     3375) + ( 930^3= 804357000) or (198^3=  7762392) + ( 927^3= 796597983) or (295^3= 25672375) + ( 920^3= 778688000)
1515  958595904 = ( 22^3=    10648) + ( 986^3= 958585256) or (180^3=  5832000) + ( 984^3= 952763904) or (692^3=331373888) + ( 856^3= 627222016)
1660 1148834232 = (222^3= 10941048) + (1044^3=1137893184) or (718^3=370146232) + ( 920^3= 778688000) or (816^3=543338496) + ( 846^3= 605495736)
1837 1407672000 = (140^3=  2744000) + (1120^3=1404928000) or (396^3= 62099136) + (1104^3=1345572864) or (630^3=250047000) + (1050^3=1157625000)
2000 1671816384 = (428^3= 78402752) + (1168^3=1593413632) or (940^3=830584000) + ( 944^3= 841232384)
2001 1672470592 = ( 29^3=    24389) + (1187^3=1672446203) or (632^3=252435968) + (1124^3=1420034624)
2002 1673170856 = (458^3= 96071912) + (1164^3=1577098944) or (828^3=567663552) + (1034^3=1105507304)
2003 1675045225 = (522^3=142236648) + (1153^3=1532808577) or (744^3=411830784) + (1081^3=1263214441)
2004 1675958167 = (492^3=119095488) + (1159^3=1556862679) or (711^3=359425431) + (1096^3=1316532736)
2005 1676926719 = ( 63^3=   250047) + (1188^3=1676676672) or (714^3=363994344) + (1095^3=1312932375)
2006 1677646971 = ( 99^3=   970299) + (1188^3=1676676672) or (891^3=707347971) + ( 990^3= 970299000)

PicoLisp

(load "@lib/simul.l")

(off 'B)
(for L (subsets 2 (range 1 1200))
   (let K (sum '((N) (** N 3)) L)
      (ifn (lup B K)
         (idx 'B (list K 1 (list L)) T)
         (inc (cdr @))
         (push (cddr @) L) ) ) )
(setq R
   (filter
      '((L) (>= (cadr L) 2))
      (idx 'B)) )
(for L (head 25 R)
   (println (car L) (caddr L)) )
(for L (head 7 (nth R 2000))
   (println (car L) (caddr L)) )
Output:
1729 ((9 10) (1 12))
4104 ((9 15) (2 16))
13832 ((18 20) (2 24))
20683 ((19 24) (10 27))
32832 ((18 30) (4 32))
39312 ((15 33) (2 34))
40033 ((16 33) (9 34))
46683 ((27 30) (3 36))
64232 ((26 36) (17 39))
65728 ((31 33) (12 40))
110656 ((36 40) (4 48))
110808 ((27 45) (6 48))
134379 ((38 43) (12 51))
149389 ((29 50) (8 53))
165464 ((38 48) (20 54))
171288 ((24 54) (17 55))
195841 ((22 57) (9 58))
216027 ((22 59) (3 60))
216125 ((45 50) (5 60))
262656 ((36 60) (8 64))
314496 ((30 66) (4 68))
320264 ((32 66) (18 68))
327763 ((51 58) (30 67))
373464 ((54 60) (6 72))
402597 ((56 61) (42 69))
1671816384 ((940 944) (428 1168))
1672470592 ((632 1124) (29 1187))
1673170856 ((828 1034) (458 1164))
1675045225 ((744 1081) (522 1153))
1675958167 ((711 1096) (492 1159))
1676926719 ((714 1095) (63 1188))
1677646971 ((891 990) (99 1188))

PureBasic

#MAX=1189

Macro q3(a,b)
  a*a*a+b*b*b
EndMacro

Structure Cap
  x.i
  y.i
  s.i
EndStructure

NewList Taxi.Cap()

For i=1 To #MAX  
  For j=i To #MAX 
    AddElement(Taxi()) : Taxi()\s=q3(i,j) : Taxi()\x=i : Taxi()\y=j 
  Next j  
Next i

SortStructuredList(Taxi(),#PB_Sort_Ascending,OffsetOf(Cap\s),TypeOf(Cap\s))

If OpenConsole()    
  ForEach Taxi()
    If sum=Taxi()\s
      r$+"="+RSet(Str(Taxi()\x),4)+"³ +"+RSet(Str(Taxi()\y),4)+"³ " : Continue
    EndIf
    If CountString(r$,"=")>=2 : c+1 : EndIf    
    If CountString(r$,"=")=2      
      Select c
        Case 1 To 25, 2000 To 2006 : PrintN(RSet(Str(c),5)+": "+RSet(Str(sum),10)+r$)        
        Case Bool(c>2006) : Break
      EndSelect        
    EndIf
    r$=""      
    sum=Taxi()\s : r$="="+RSet(Str(Taxi()\x),4)+"³ +"+RSet(Str(Taxi()\y),4)+"³ "    
  Next  
  PrintN("FIN.") : Input()
EndIf
Output:
    1:       1729=   1³ +  12³ =   9³ +  10³ 
    2:       4104=   2³ +  16³ =   9³ +  15³ 
    3:      13832=   2³ +  24³ =  18³ +  20³ 
    4:      20683=  10³ +  27³ =  19³ +  24³ 
    5:      32832=   4³ +  32³ =  18³ +  30³ 
    6:      39312=   2³ +  34³ =  15³ +  33³ 
    7:      40033=   9³ +  34³ =  16³ +  33³ 
    8:      46683=   3³ +  36³ =  27³ +  30³ 
    9:      64232=  17³ +  39³ =  26³ +  36³ 
   10:      65728=  12³ +  40³ =  31³ +  33³ 
   11:     110656=   4³ +  48³ =  36³ +  40³ 
   12:     110808=   6³ +  48³ =  27³ +  45³ 
   13:     134379=  12³ +  51³ =  38³ +  43³ 
   14:     149389=   8³ +  53³ =  29³ +  50³ 
   15:     165464=  20³ +  54³ =  38³ +  48³ 
   16:     171288=  17³ +  55³ =  24³ +  54³ 
   17:     195841=   9³ +  58³ =  22³ +  57³ 
   18:     216027=   3³ +  60³ =  22³ +  59³ 
   19:     216125=   5³ +  60³ =  45³ +  50³ 
   20:     262656=   8³ +  64³ =  36³ +  60³ 
   21:     314496=   4³ +  68³ =  30³ +  66³ 
   22:     320264=  18³ +  68³ =  32³ +  66³ 
   23:     327763=  30³ +  67³ =  51³ +  58³ 
   24:     373464=   6³ +  72³ =  54³ +  60³ 
   25:     402597=  42³ +  69³ =  56³ +  61³ 
 2000: 1671816384= 428³ +1168³ = 940³ + 944³ 
 2001: 1672470592=  29³ +1187³ = 632³ +1124³ 
 2002: 1673170856= 458³ +1164³ = 828³ +1034³ 
 2003: 1675045225= 522³ +1153³ = 744³ +1081³ 
 2004: 1675958167= 492³ +1159³ = 711³ +1096³ 
 2005: 1676926719=  63³ +1188³ = 714³ +1095³ 
 2006: 1677646971=  99³ +1188³ = 891³ + 990³ 
FIN.

Python

(Magic number 1201 found by trial and error)

from collections import defaultdict
from itertools import product
from pprint import pprint as pp

cube2n = {x**3:x for x in range(1, 1201)}
sum2cubes = defaultdict(set)
for c1, c2 in product(cube2n, cube2n):
	if c1 >= c2: sum2cubes[c1 + c2].add((cube2n[c1], cube2n[c2]))
	
taxied = sorted((k, v) for k,v in sum2cubes.items() if len(v) >= 2)

#pp(len(taxied))  # 2068
for t in enumerate(taxied[:25], 1):
    pp(t)
print('...')    
for t in enumerate(taxied[2000-1:2000+6], 2000):
    pp(t)
Output:
(1, (1729, {(12, 1), (10, 9)}))
(2, (4104, {(16, 2), (15, 9)}))
(3, (13832, {(20, 18), (24, 2)}))
(4, (20683, {(27, 10), (24, 19)}))
(5, (32832, {(30, 18), (32, 4)}))
(6, (39312, {(33, 15), (34, 2)}))
(7, (40033, {(33, 16), (34, 9)}))
(8, (46683, {(30, 27), (36, 3)}))
(9, (64232, {(36, 26), (39, 17)}))
(10, (65728, {(33, 31), (40, 12)}))
(11, (110656, {(48, 4), (40, 36)}))
(12, (110808, {(48, 6), (45, 27)}))
(13, (134379, {(51, 12), (43, 38)}))
(14, (149389, {(50, 29), (53, 8)}))
(15, (165464, {(54, 20), (48, 38)}))
(16, (171288, {(54, 24), (55, 17)}))
(17, (195841, {(57, 22), (58, 9)}))
(18, (216027, {(60, 3), (59, 22)}))
(19, (216125, {(60, 5), (50, 45)}))
(20, (262656, {(64, 8), (60, 36)}))
(21, (314496, {(66, 30), (68, 4)}))
(22, (320264, {(66, 32), (68, 18)}))
(23, (327763, {(58, 51), (67, 30)}))
(24, (373464, {(72, 6), (60, 54)}))
(25, (402597, {(69, 42), (61, 56)}))
...
(2000, (1671816384, {(1168, 428), (944, 940)}))
(2001, (1672470592, {(1187, 29), (1124, 632)}))
(2002, (1673170856, {(1164, 458), (1034, 828)}))
(2003, (1675045225, {(1153, 522), (1081, 744)}))
(2004, (1675958167, {(1159, 492), (1096, 711)}))
(2005, (1676926719, {(1188, 63), (1095, 714)}))
(2006, (1677646971, {(990, 891), (1188, 99)}))

Although, for this task it's simply faster to look up the cubes in the sum when we need to print them, because we can now store and sort only the sums:

cubes, crev = [x**3 for x in range(1,1200)], {}
# for cube root lookup
for x,x3 in enumerate(cubes): crev[x3] = x + 1

sums = sorted(x+y for x in cubes for y in cubes if y < x)

idx = 0
for i in range(1, len(sums)-1):
    if sums[i-1] != sums[i] and sums[i] == sums[i+1]:
        idx += 1
        if idx > 25 and idx < 2000 or idx > 2006: continue

        n,p = sums[i],[]
        for x in cubes:
            if n-x < x: break
            if n-x in crev:
                p.append((crev[x], crev[n-x]))
        print "%4d: %10d"%(idx,n),
        for x in p: print " = %4d^3 + %4d^3"%x,
        print
Output:

Output trimmed to reduce clutter.

   1:       1729  =    1^3 +   12^3  =    9^3 +   10^3
   2:       4104  =    2^3 +   16^3  =    9^3 +   15^3
   3:      13832  =    2^3 +   24^3  =   18^3 +   20^3
   4:      20683  =   10^3 +   27^3  =   19^3 +   24^3
   5:      32832  =    4^3 +   32^3  =   18^3 +   30^3
...
2004: 1675958167  =  492^3 + 1159^3  =  711^3 + 1096^3
2005: 1676926719  =   63^3 + 1188^3  =  714^3 + 1095^3
2006: 1677646971  =   99^3 + 1188^3  =  891^3 +  990^3

Using heapq module

A priority queue that holds cube sums. When consecutive sums come out with the same value, they are taxis.

from heapq import heappush, heappop

def cubesum():
    h,n = [],1
    while True:
        while not h or h[0][0] > n**3: # could also pre-calculate cubes
            heappush(h, (n**3 + 1, n, 1))
            n += 1

        (s, x, y) = heappop(h)
        yield((s, x, y))
        y += 1
        if y < x:    # should be y <= x?
            heappush(h, (x**3 + y**3, x, y))

def taxis():
    out = [(0,0,0)]
    for s in cubesum():
        if s[0] == out[-1][0]:
            out.append(s)
        else:
            if len(out) > 1: yield(out)
            out = [s]

n = 0
for x in taxis():
    n += 1
    if n >= 2006: break
    if n <= 25 or n >= 2000:
        print(n, x)
Output:
(1, [(1729, 10, 9), (1729, 12, 1)])
(2, [(4104, 15, 9), (4104, 16, 2)])
(3, [(13832, 20, 18), (13832, 24, 2)])
(4, [(20683, 24, 19), (20683, 27, 10)])
(5, [(32832, 30, 18), (32832, 32, 4)])
(6, [(39312, 33, 15), (39312, 34, 2)])
(7, [(40033, 33, 16), (40033, 34, 9)])
(8, [(46683, 30, 27), (46683, 36, 3)])
(9, [(64232, 36, 26), (64232, 39, 17)])
(10, [(65728, 33, 31), (65728, 40, 12)])
(11, [(110656, 40, 36), (110656, 48, 4)])
(12, [(110808, 45, 27), (110808, 48, 6)])
(13, [(134379, 43, 38), (134379, 51, 12)])
(14, [(149389, 50, 29), (149389, 53, 8)])
(15, [(165464, 48, 38), (165464, 54, 20)])
(16, [(171288, 54, 24), (171288, 55, 17)])
(17, [(195841, 57, 22), (195841, 58, 9)])
(18, [(216027, 59, 22), (216027, 60, 3)])
(19, [(216125, 50, 45), (216125, 60, 5)])
(20, [(262656, 60, 36), (262656, 64, 8)])
(21, [(314496, 66, 30), (314496, 68, 4)])
(22, [(320264, 66, 32), (320264, 68, 18)])
(23, [(327763, 58, 51), (327763, 67, 30)])
(24, [(373464, 60, 54), (373464, 72, 6)])
(25, [(402597, 61, 56), (402597, 69, 42)])
(2000, [(1671816384, 944, 940), (1671816384, 1168, 428)])
(2001, [(1672470592, 1124, 632), (1672470592, 1187, 29)])
(2002, [(1673170856, 1034, 828), (1673170856, 1164, 458)])
(2003, [(1675045225, 1081, 744), (1675045225, 1153, 522)])
(2004, [(1675958167, 1096, 711), (1675958167, 1159, 492)])
(2005, [(1676926719, 1095, 714), (1676926719, 1188, 63)])

Racket

This is the straighforward implementation, so it finds only the first 25 values in a sensible amount of time.

#lang racket

(define (cube x) (* x x x))

;floor of cubic root
(define (cubic-root x)
  (let ([aprox (inexact->exact (round (expt x (/ 1 3))))])
    (if (> (cube aprox) x)
        (- aprox 1)
        aprox)))

(let loop ([p 1] [n 1])
  (let ()
    (define pairs
      (for*/list ([j (in-range 1 (add1 (cubic-root (quotient n 2))))]
                  [k (in-value (cubic-root (- n (cube j))))]
                  #:when (= n (+ (cube j) (cube k))))
        (cons j k)))
    (if (>= (length pairs) 2)
      (begin
        (printf "~a: ~a" p n)
        (for ([pair (in-list pairs)])
          (printf " = ~a^3 + ~a^3" (car pair) (cdr pair)))
          (newline)
        (when (< p 25)
          (loop (add1 p) (add1 n))))
      (loop p (add1 n)))))
Output:
1: 1729 = 1^3 + 12^3 = 9^3 + 10^3
2: 4104 = 2^3 + 16^3 = 9^3 + 15^3
3: 13832 = 2^3 + 24^3 = 18^3 + 20^3
4: 20683 = 10^3 + 27^3 = 19^3 + 24^3
5: 32832 = 4^3 + 32^3 = 18^3 + 30^3
6: 39312 = 2^3 + 34^3 = 15^3 + 33^3
7: 40033 = 9^3 + 34^3 = 16^3 + 33^3
8: 46683 = 3^3 + 36^3 = 27^3 + 30^3
9: 64232 = 17^3 + 39^3 = 26^3 + 36^3
10: 65728 = 12^3 + 40^3 = 31^3 + 33^3
11: 110656 = 4^3 + 48^3 = 36^3 + 40^3
12: 110808 = 6^3 + 48^3 = 27^3 + 45^3
13: 134379 = 12^3 + 51^3 = 38^3 + 43^3
14: 149389 = 8^3 + 53^3 = 29^3 + 50^3
15: 165464 = 20^3 + 54^3 = 38^3 + 48^3
16: 171288 = 17^3 + 55^3 = 24^3 + 54^3
17: 195841 = 9^3 + 58^3 = 22^3 + 57^3
18: 216027 = 3^3 + 60^3 = 22^3 + 59^3
19: 216125 = 5^3 + 60^3 = 45^3 + 50^3
20: 262656 = 8^3 + 64^3 = 36^3 + 60^3
21: 314496 = 4^3 + 68^3 = 30^3 + 66^3
22: 320264 = 18^3 + 68^3 = 32^3 + 66^3
23: 327763 = 30^3 + 67^3 = 51^3 + 58^3
24: 373464 = 6^3 + 72^3 = 54^3 + 60^3
25: 402597 = 42^3 + 69^3 = 56^3 + 61^3

Raku

(formerly Perl 6)

This uses a pretty simple search algorithm that doesn't necessarily return the Taxicab numbers in order. Assuming we want all the Taxicab numbers within some range S to N, we'll search until we find N values. When we find the Nth value, we continue to search up to the cube root of the largest Taxicab number found up to that point. That ensures we will find all of them inside the desired range without needing to search arbitrarily or use magic numbers. Defaults to returning the Taxicab numbers from 1 to 25. Pass in a different start and end value if you want some other range.

constant @cu = (^Inf).map: { .³ }

sub MAIN ($start = 1, $end = 25) {
    my %taxi;
    my int $taxis = 0;
    my $terminate = 0;
    my int $max = 0;

    for 1 .. * -> $c1 {
        last if ?$terminate && ($terminate < $c1);
        for 1 .. $c1 -> $c2 {
            my $this = @cu[$c1] + @cu[$c2];
            %taxi{$this}.push: [$c2, $c1];
            if %taxi{$this}.elems == 2 {
                ++$taxis;
                $max max= $this;
            }
    	    $terminate = ceiling $max ** (1/3) if $taxis == $end and !$terminate;
        }   
    }

    display( %taxi, $start, $end );

}

sub display (%this_stuff, $start, $end) {
    my $i = $start;
    printf "%4d %10d  =>\t%s\n", $i++, $_.key,
        (.value.map({ sprintf "%4d³ + %-s\³", |$_ })).join: ",\t"
        for %this_stuff.grep( { $_.value.elems > 1 } ).sort( +*.key )[$start-1..$end-1];
}
Output:

With no passed parameters (default)

   1       1729  =>	   9³ + 10³,	   1³ + 12³
   2       4104  =>	   9³ + 15³,	   2³ + 16³
   3      13832  =>	  18³ + 20³,	   2³ + 24³
   4      20683  =>	  19³ + 24³,	  10³ + 27³
   5      32832  =>	  18³ + 30³,	   4³ + 32³
   6      39312  =>	  15³ + 33³,	   2³ + 34³
   7      40033  =>	  16³ + 33³,	   9³ + 34³
   8      46683  =>	  27³ + 30³,	   3³ + 36³
   9      64232  =>	  26³ + 36³,	  17³ + 39³
  10      65728  =>	  31³ + 33³,	  12³ + 40³
  11     110656  =>	  36³ + 40³,	   4³ + 48³
  12     110808  =>	  27³ + 45³,	   6³ + 48³
  13     134379  =>	  38³ + 43³,	  12³ + 51³
  14     149389  =>	  29³ + 50³,	   8³ + 53³
  15     165464  =>	  38³ + 48³,	  20³ + 54³
  16     171288  =>	  24³ + 54³,	  17³ + 55³
  17     195841  =>	  22³ + 57³,	   9³ + 58³
  18     216027  =>	  22³ + 59³,	   3³ + 60³
  19     216125  =>	  45³ + 50³,	   5³ + 60³
  20     262656  =>	  36³ + 60³,	   8³ + 64³
  21     314496  =>	  30³ + 66³,	   4³ + 68³
  22     320264  =>	  32³ + 66³,	  18³ + 68³
  23     327763  =>	  51³ + 58³,	  30³ + 67³
  24     373464  =>	  54³ + 60³,	   6³ + 72³
  25     402597  =>	  56³ + 61³,	  42³ + 69³

With passed parameters 2000 2006:

2000 1671816384  =>	 940³ + 944³,	 428³ + 1168³
2001 1672470592  =>	 632³ + 1124³,	  29³ + 1187³
2002 1673170856  =>	 828³ + 1034³,	 458³ + 1164³
2003 1675045225  =>	 744³ + 1081³,	 522³ + 1153³
2004 1675958167  =>	 711³ + 1096³,	 492³ + 1159³
2005 1676926719  =>	 714³ + 1095³,	  63³ + 1188³
2006 1677646971  =>	 891³ + 990³,	  99³ + 1188³

REXX

Programming note:   to ensure that the taxicab numbers are in order, an extra 10% are generated.

/*REXX program displays the specified first (lowest) taxicab numbers (for three ranges).*/
parse arg  L.1  H.1      L.2 H.2      L.3 H.3 .  /*obtain optional arguments from the CL*/
   if L.1=='' | L.1==","  then L.1=   1          /*L1  is the low  part of 1st range.   */
   if H.1=='' | H.1==","  then H.1=  25          /*H1   "  "  high   "   "  "    "      */
   if L.2=='' | L.2==","  then L.2= 454          /*L2   "  "  low    "   " 2nd   "      */
   if H.2=='' | H.2==","  then H.2= 456          /*H2   "  "  high   "   "  "    "      */
   if L.3=='' | L.3==","  then L.3=2000          /*L3   "  "  low    "   " 3rd   "      */
   if H.3=='' | H.3==","  then H.3=2006          /*H3   "  "  high   "   "  "    "      */
mx= max(H.1,    H.2,    H.3)                     /*find how many taxicab numbers needed.*/
mx= mx   + mx % 10                               /*cushion;  compensate for the triples.*/
ww= length(mx) * 3;           w= ww % 2          /*widths used for formatting the output*/
numeric digits max(9, ww)                        /*prepare to use some larger numbers.  */
@.=.;    #= 0;     @@.= 0;    @and= "  ──and── " /*set some REXX vars and handy literals*/
$.=                                              /* [↓]  generate extra taxicab numbers.*/
    do j=1  until #>=mx;            C= j**3      /*taxicab numbers may not be in order. */
    !.j= C                                       /*use memoization for cube calculation.*/
      do k=1  for j-1;              s= C + !.k   /*define a whole bunch of cube sums.   */
      if @.s==.  then do;  @.s= j;  b.s= k       /*Cube not defined?   Then process it. */
                           iterate               /*define  @.S  and  B.S≡sum  of 2 cubes*/
                      end                        /* [↑]  define one cube sum at a time. */
      has= @@.s                                  /*has this number been defined before? */
      if has  then $.s= $.s @and U(j,'   +')U(k) /* ◄─ build a display string. [↓]      */
              else $.s= right(s,ww)  '───►'   U(@.s,"   +")U(b.s)   @and   U(j,'   +')U(k)
      @@.s= 1                                    /*mark taxicab number as a sum of cubes*/
      if has   then iterate                      /*S  is a triple (or sometimes better).*/
      #= # + 1;    #.#= s                        /*bump taxicab counter; define taxicab#*/
      end   /*k*/                                /* [↑]  build the cubes one─at─a─time. */
    end      /*j*/                               /* [↑]  complete with overage numbers. */
A.=
       do k=1  for mx;   _= #.k;    A.k= $._     /*re─assign disjoint $. elements to A. */
       end   /*k*/
call Esort  mx                                   /*sort taxicab #s with an exchange sort*/
       do grp=1  for 3;  call tell L.grp, H.grp  /*display the three grps of numbers. */
       end   /*grp*/
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
tell:  do t=arg(1)  to arg(2);    say right(t, 9)':'   A.t;     end;        say;    return
U:     return right(arg(1), w)'^3'arg(2)         /*right─justify a number,  append "^3" */
/*──────────────────────────────────────────────────────────────────────────────────────*/
Esort: procedure expose A.; parse arg N;         h= N    /*Esort when items have blanks.*/
         do  while h>1;     h= h % 2
           do i=1  for N-h;           k=h + i;   j= i
              do forever;   parse var A.k xk .;  parse var A.j xj .;  if xk>=xj then leave
              _= A.j;       A.j= A.k; A.k= _             /*swap two elements of A. array*/
              if h>=j  then leave;    j=j - h;   k= k - h
              end   /*forever*/
           end      /*i*/
         end        /*while h>1*/;               return
output   when using the default inputs:
 
        1:         1729 ───►     10^3   +     9^3   ──and──      12^3   +     1^3
        2:         4104 ───►     15^3   +     9^3   ──and──      16^3   +     2^3
        3:        13832 ───►     20^3   +    18^3   ──and──      24^3   +     2^3
        4:        20683 ───►     24^3   +    19^3   ──and──      27^3   +    10^3
        5:        32832 ───►     30^3   +    18^3   ──and──      32^3   +     4^3
        6:        39312 ───►     33^3   +    15^3   ──and──      34^3   +     2^3
        7:        40033 ───►     33^3   +    16^3   ──and──      34^3   +     9^3
        8:        46683 ───►     30^3   +    27^3   ──and──      36^3   +     3^3
        9:        64232 ───►     36^3   +    26^3   ──and──      39^3   +    17^3
       10:        65728 ───►     33^3   +    31^3   ──and──      40^3   +    12^3
       11:       110656 ───►     40^3   +    36^3   ──and──      48^3   +     4^3
       12:       110808 ───►     45^3   +    27^3   ──and──      48^3   +     6^3
       13:       134379 ───►     43^3   +    38^3   ──and──      51^3   +    12^3
       14:       149389 ───►     50^3   +    29^3   ──and──      53^3   +     8^3
       15:       165464 ───►     48^3   +    38^3   ──and──      54^3   +    20^3
       16:       171288 ───►     54^3   +    24^3   ──and──      55^3   +    17^3
       17:       195841 ───►     57^3   +    22^3   ──and──      58^3   +     9^3
       18:       216027 ───►     59^3   +    22^3   ──and──      60^3   +     3^3
       19:       216125 ───►     50^3   +    45^3   ──and──      60^3   +     5^3
       20:       262656 ───►     60^3   +    36^3   ──and──      64^3   +     8^3
       21:       314496 ───►     66^3   +    30^3   ──and──      68^3   +     4^3
       22:       320264 ───►     66^3   +    32^3   ──and──      68^3   +    18^3
       23:       327763 ───►     58^3   +    51^3   ──and──      67^3   +    30^3
       24:       373464 ───►     60^3   +    54^3   ──and──      72^3   +     6^3
       25:       402597 ───►     61^3   +    56^3   ──and──      69^3   +    42^3

      454:     87483968 ───►    363^3   +   341^3   ──and──     440^3   +   132^3
      455:     87539319 ───►    414^3   +   255^3   ──and──     423^3   +   228^3   ──and──     436^3   +   167^3
      456:     87579037 ───►    370^3   +   333^3   ──and──     444^3   +    37^3

     2000:   1671816384 ───►    944^3   +   940^3   ──and──    1168^3   +   428^3
     2001:   1672470592 ───►   1124^3   +   632^3   ──and──    1187^3   +    29^3
     2002:   1673170856 ───►   1034^3   +   828^3   ──and──    1164^3   +   458^3
     2003:   1675045225 ───►   1081^3   +   744^3   ──and──    1153^3   +   522^3
     2004:   1675958167 ───►   1096^3   +   711^3   ──and──    1159^3   +   492^3
     2005:   1676926719 ───►   1095^3   +   714^3   ──and──    1188^3   +    63^3
     2006:   1677646971 ───►    990^3   +   891^3   ──and──    1188^3   +    99^3

Ring

# Project : Taxicab numbers

num = 0
for n = 1 to 500000
    nr = 0
    tax = []
    for m = 1 to 75
        for p = m + 1 to 75
            if n = pow(m, 3) + pow(p, 3)
               add(tax, m)
               add(tax, p)
               nr = nr + 1
            ok
        next
    next
    if nr > 1
       num = num + 1
       see "" + num + " " + n + " => " + tax[1] + "^3 + " + tax[2] + "^3" + ", "
       see "" + tax[3] + "^3 +" + tax[4] + "^3" + nl
       if num = 25
          exit
       ok
    ok
next
see "ok" + nl

Output:

   1       1729  =>	   9³ + 10³,	   1³ + 12³
   2       4104  =>	   9³ + 15³,	   2³ + 16³
   3      13832  =>	  18³ + 20³,	   2³ + 24³
   4      20683  =>	  19³ + 24³,	  10³ + 27³
   5      32832  =>	  18³ + 30³,	   4³ + 32³
   6      39312  =>	  15³ + 33³,	   2³ + 34³
   7      40033  =>	  16³ + 33³,	   9³ + 34³
   8      46683  =>	  27³ + 30³,	   3³ + 36³
   9      64232  =>	  26³ + 36³,	  17³ + 39³
  10     65728  =>	  31³ + 33³,	  12³ + 40³
  11     110656  =>	  36³ + 40³,	   4³ + 48³
  12     110808  =>	  27³ + 45³,	   6³ + 48³
  13     134379  =>	  38³ + 43³,	  12³ + 51³
  14     149389  =>	  29³ + 50³,	   8³ + 53³
  15     165464  =>	  38³ + 48³,	  20³ + 54³
  16     171288  =>	  24³ + 54³,	  17³ + 55³
  17     195841  =>	  22³ + 57³,	   9³ + 58³
  18     216027  =>	  22³ + 59³,	   3³ + 60³
  19     216125  =>	  45³ + 50³,	   5³ + 60³
  20     262656  =>	  36³ + 60³,	   8³ + 64³
  21     314496  =>	  30³ + 66³,	   4³ + 68³
  22     320264  =>	  32³ + 66³,	  18³ + 68³
  23     327763  =>	  51³ + 58³,	  30³ + 67³
  24     373464  =>	  54³ + 60³,	   6³ + 72³
  25     402597  =>	  56³ + 61³,	  42³ + 69³
  ok

RPL

A priority queue allows to get the first 25th taxicab numbers without being killed by the emulator timedog.

Works with: Halcyon Calc version 4.2.7
Code Comments
≪ 
   {} 1 PQueue SIZE FOR j
        IF OVER j ≠ THEN PQueue j GET 1 →LIST + END
     NEXT
     'PQueue' STO DROP
≫ 'PQPop' STO

≪ 
  0 {} → idx item 
  ≪ 0 1 PQueue SIZE FOR j
        PQueue j GET 1 GET
        IF DUP2 < 
        THEN SWAP PQueue j GET 'item' STO j 'idx' STO 
       END DROP
     NEXT
     DROP idx item
≫ ≫ 'PQMax' STO

≪ → pos
  1 CF PQueue SIZE WHILE DUP 1 FC? AND REPEAT
     IF PQueue OVER GET pos GET 3 PICK == THEN 1 SF END
     1 -
     END
  DROP2 1 FS? 
≫ ≫ 'PQin?' STO

≪ OVER 3 ^ OVER 3 ^ + ROT ROT 
 3 →LIST 1 →LIST PQueue + 'PQueue' STO 
≫ 'PQAdd' STO

≪ → item
  ≪ IF item 1 GET LastItem 1 GET == THEN
        {} item 1 GET + item 2 GET item 3 GET R→C +
        LastItem 2 GET LastItem 3 GET R→C + 1 →LIST
        TaxiNums + 'TaxiNums' STO
     END
     item 'LastItem' STO
≫ ≫  'StoreIfTwice' STO

≪ WHILE PQueue SIZE REPEAT
     PQMax DUP StoreIfTwice LIST→ DROP 4 ROLL PQPop → r c
     ≪ IF r 1 > 
       THEN IF r 1 - 2 PQin? NOT 
             THEN r 1 - c PQAdd END END
        IF c 1 > c r > AND 
       THEN IF c 1 - 3 PQin? NOT 
             THEN r c 1 - PQAdd END
        END
       DROP
     ≫ 
  END
≫ 'TAXI' STO

≪ → n 
  ≪ { 0 0 0 } 'LastItem' STO 
  { } DUP 'PQueue' STO 'TaxiNums' STO n n PQAdd 
≫ ≫ 'INIT 'STO
( j --  )
Scan the priority queue
Copy all items except one 




(  -- index { PQ_item } )
Look for PQ item with max value









( x pos -- boolean )







( r, c -- { PQ_item } )



( x pos -- boolean )
PQ item = { r^3+c^3 r c }
 






(  --  )
Removing max value from queue - and store it if déjà vu

can we add an item from the row above?

can we add an item from the left column and above diagonal?








( magic_number -- )
initialize global variables

The following commands deliver what is required:

70 INIT TAXI
TaxiNums
Output:
1: { { 1729 (9,10) (1,12) } { 4104 (9,15) (2,16) } { 13832 (18,20) (2,24) } { 20683 (19,24) (10,27) } { 32832 (18,30) (4,32) } { 39312 (15,33) (2,34) } { 40033 (16,33) (9,34) } { 46683 (27,30) (3,36) } { 64232 (26,36) (17,39) } { 65728 (31,33) (12,40) } { 110656 (36,40) (4,48) } { 110808 (27,45) (6,48) } { 134379 (38,43) (12,51) } { 149389 (29,50) (8,53) } { 165464 (38,48) (20,54) } { 171288 (24,54) (17,55) } { 195841 (22,57) (9,58) } { 216027 (22,59) (3,60) } { 216125 (45,50) (5,60) } { 262656 (36,60) (8,64) } { 314496 (30,66) (4,68) } { 320264 (32,66) (18,68) } { 327763 (51,58) (30,67) } { 402597 (56,61) (42,69) } }

Ruby

def taxicab_number(nmax=1200)
  [*1..nmax].repeated_combination(2).group_by{|x,y| x**3 + y**3}.select{|k,v| v.size>1}.sort
end

t = [0] + taxicab_number

[*1..25, *2000...2007].each do |i|
  puts "%4d: %10d" % [i, t[i][0]] + t[i][1].map{|a| " = %4d**3 + %4d**3" % a}.join
end
Output:
   1:       1729 =    1**3 +   12**3 =    9**3 +   10**3
   2:       4104 =    2**3 +   16**3 =    9**3 +   15**3
   3:      13832 =    2**3 +   24**3 =   18**3 +   20**3
   4:      20683 =   10**3 +   27**3 =   19**3 +   24**3
   5:      32832 =    4**3 +   32**3 =   18**3 +   30**3
   6:      39312 =    2**3 +   34**3 =   15**3 +   33**3
   7:      40033 =    9**3 +   34**3 =   16**3 +   33**3
   8:      46683 =    3**3 +   36**3 =   27**3 +   30**3
   9:      64232 =   17**3 +   39**3 =   26**3 +   36**3
  10:      65728 =   12**3 +   40**3 =   31**3 +   33**3
  11:     110656 =    4**3 +   48**3 =   36**3 +   40**3
  12:     110808 =    6**3 +   48**3 =   27**3 +   45**3
  13:     134379 =   12**3 +   51**3 =   38**3 +   43**3
  14:     149389 =    8**3 +   53**3 =   29**3 +   50**3
  15:     165464 =   20**3 +   54**3 =   38**3 +   48**3
  16:     171288 =   17**3 +   55**3 =   24**3 +   54**3
  17:     195841 =    9**3 +   58**3 =   22**3 +   57**3
  18:     216027 =    3**3 +   60**3 =   22**3 +   59**3
  19:     216125 =    5**3 +   60**3 =   45**3 +   50**3
  20:     262656 =    8**3 +   64**3 =   36**3 +   60**3
  21:     314496 =    4**3 +   68**3 =   30**3 +   66**3
  22:     320264 =   18**3 +   68**3 =   32**3 +   66**3
  23:     327763 =   30**3 +   67**3 =   51**3 +   58**3
  24:     373464 =    6**3 +   72**3 =   54**3 +   60**3
  25:     402597 =   42**3 +   69**3 =   56**3 +   61**3
2000: 1671816384 =  428**3 + 1168**3 =  940**3 +  944**3
2001: 1672470592 =   29**3 + 1187**3 =  632**3 + 1124**3
2002: 1673170856 =  458**3 + 1164**3 =  828**3 + 1034**3
2003: 1675045225 =  522**3 + 1153**3 =  744**3 + 1081**3
2004: 1675958167 =  492**3 + 1159**3 =  711**3 + 1096**3
2005: 1676926719 =   63**3 + 1188**3 =  714**3 + 1095**3
2006: 1677646971 =   99**3 + 1188**3 =  891**3 +  990**3

Rust

use std::collections::HashMap;
use itertools::Itertools;

fn cubes(n: u64) -> Vec<u64> {
	let mut cube_vector = Vec::new();
	for i in 1..=n {
		cube_vector.push(i.pow(3));
	}
	cube_vector
}

fn main() {
	let c = cubes(1201);
	let it = c.iter().combinations(2);
	let mut m = HashMap::new();
	for x in it {
		let sum = x[0] + x[1];
		m.entry(sum).or_insert(Vec::new()).push(x)
	} 
	
	let mut result = Vec::new();
	
	for (k,v) in m.iter() {
		if v.len() > 1 {
			result.push((k,v));
		}
	}
	
	result.sort();
	for f in result {
		println!("{:?}", f);
	}
}
Output:

(1729, [[1, 1728], [729, 1000]])
(4104, [[8, 4096], [729, 3375]])
(13832, [[8, 13824], [5832, 8000]])
(20683, [[1000, 19683], [6859, 13824]])
(32832, [[64, 32768], [5832, 27000]])
(39312, [[8, 39304], [3375, 35937]])
(40033, [[729, 39304], [4096, 35937]])
(46683, [[27, 46656], [19683, 27000]])
(64232, [[4913, 59319], [17576, 46656]])
(65728, [[1728, 64000], [29791, 35937]])
(110656, [[64, 110592], [46656, 64000]])
(110808, [[216, 110592], [19683, 91125]])
(134379, [[1728, 132651], [54872, 79507]])
(149389, [[512, 148877], [24389, 125000]])
(165464, [[8000, 157464], [54872, 110592]])
(171288, [[4913, 166375], [13824, 157464]])
(195841, [[729, 195112], [10648, 185193]])
(216027, [[27, 216000], [10648, 205379]])
(216125, [[125, 216000], [91125, 125000]])
(262656, [[512, 262144], [46656, 216000]])
(314496, [[64, 314432], [27000, 287496]])

Scala

import scala.collection.MapView
import scala.math.pow

implicit class Pairs[A, B]( p:List[(A, B)]) {
  def collectPairs: MapView[A, List[B]] = p.groupBy(_._1).view.mapValues(_.map(_._2)).filterNot(_._2.size<2)
}

// Make a sorted List of Taxi Cab Numbers. Limit it to the cube of 1200 because we know it's high enough.
val taxiNums = {
  (1 to 1200).toList            // Start with a sequential list of integers
    .combinations(2).toList     // Find all two number combinations
    .map {
      case a :: b :: nil => ((pow(a, 3) + pow(b, 3)).toInt, (a, b))
      case _ => 0 ->(0, 0)
    }                           // Turn the list into the sum of two cubes and
    //      remember what we started with, eg. 28->(1,3)
    .collectPairs               // Only keep taxi cab numbers with a duplicate
    .toList.sortBy(_._1)        // Sort the results
}

def output() : Unit = {
  println( "%20s".format( "Taxi Cab Numbers" ) )
  println( "%20s%15s%15s".format( "-"*20, "-"*15, "-"*15 ) )

  taxiNums.take(25) foreach {
    case (p, a::b::Nil) => println( "%20d\t(%d\u00b3 + %d\u00b3)\t\t(%d\u00b3 + %d\u00b3)".format(p,a._1,a._2,b._1,b._2) )
  }

  taxiNums.slice(1999,2007) foreach {
    case (p, a::b::Nil) => println( "%20d\t(%d\u00b3 + %d\u00b3)\t(%d\u00b3 + %d\u00b3)".format(p,a._1,a._2,b._1,b._2) )
  }
}
Output:
    Taxi Cab Numbers
--------------------------------------------------
                1729	(1³ + 12³)	(9³ + 10³)
                4104	(2³ + 16³)	(9³ + 15³)
               13832	(2³ + 24³)	(18³ + 20³)
               20683	(10³ + 27³)	(19³ + 24³)
               32832	(4³ + 32³)	(18³ + 30³)
               39312	(2³ + 34³)	(15³ + 33³)
               40033	(9³ + 34³)	(16³ + 33³)
               46683	(3³ + 36³)	(27³ + 30³)
               64232	(17³ + 39³)	(26³ + 36³)
               65728	(12³ + 40³)	(31³ + 33³)
              110656	(4³ + 48³)	(36³ + 40³)
              110808	(6³ + 48³)	(27³ + 45³)
              134379	(12³ + 51³)	(38³ + 43³)
              149389	(8³ + 53³)	(29³ + 50³)
              165464	(20³ + 54³)	(38³ + 48³)
              171288	(17³ + 55³)	(24³ + 54³)
              195841	(9³ + 58³)	(22³ + 57³)
              216027	(3³ + 60³)	(22³ + 59³)
              216125	(5³ + 60³)	(45³ + 50³)
              262656	(8³ + 64³)	(36³ + 60³)
              314496	(4³ + 68³)	(30³ + 66³)
              320264	(18³ + 68³)	(32³ + 66³)
              327763	(30³ + 67³)	(51³ + 58³)
              373464	(6³ + 72³)	(54³ + 60³)
              402597	(42³ + 69³)	(56³ + 61³)
          1671816384	(428³ + 1168³)	(940³ + 944³)
          1672470592	(29³ + 1187³)	(632³ + 1124³)
          1673170856	(458³ + 1164³)	(828³ + 1034³)
          1675045225	(522³ + 1153³)	(744³ + 1081³)
          1675958167	(492³ + 1159³)	(711³ + 1096³)
          1676926719	(63³ + 1188³)	(714³ + 1095³)
          1677646971	(99³ + 1188³)	(891³ + 990³)

Scheme

Library: Scheme/SRFIs
(import (scheme base)
        (scheme write)
        (srfi 1)        ; lists
        (srfi 69)       ; hash tables
        (srfi 132))     ; sorting

(define *max-n* 1500) ; let's go up to here, maximum for x and y
(define *numbers* (make-hash-table eqv?)) ; hash table for total -> list of list of pairs

(define (retrieve key) (hash-table-ref/default *numbers* key '()))

;; add all combinations to the hash table
(do ((i 1 (+ i 1)))
  ((= i *max-n*) )
  (do ((j (+ 1 i) (+ j 1)))
    ((= j *max-n*) )
    (let ((n (+ (* i i i) (* j j j))))
      (hash-table-set! *numbers* n 
                       (cons (list i j) (retrieve n))))))

(define (display-number i key)
  (display (+ 1 i)) (display ": ") 
  (display key) (display " -> ")
  (display (retrieve key)) (newline))
  
(let ((sorted-keys (list-sort < 
                              (filter (lambda (key) (> (length (retrieve key)) 1))
                                      (hash-table-keys *numbers*)))))
  ;; first 25 
  (for-each (lambda (i) (display-number i (list-ref sorted-keys i)))
            (iota 25))
  ;; 2000-2006
  (for-each (lambda (i) (display-number i (list-ref sorted-keys i)))
            (iota 7 1999))
  )
Output:
1: 1729 -> ((9 10) (1 12))
2: 4104 -> ((9 15) (2 16))
3: 13832 -> ((18 20) (2 24))
4: 20683 -> ((19 24) (10 27))
5: 32832 -> ((18 30) (4 32))
6: 39312 -> ((15 33) (2 34))
7: 40033 -> ((16 33) (9 34))
8: 46683 -> ((27 30) (3 36))
9: 64232 -> ((26 36) (17 39))
10: 65728 -> ((31 33) (12 40))
11: 110656 -> ((36 40) (4 48))
12: 110808 -> ((27 45) (6 48))
13: 134379 -> ((38 43) (12 51))
14: 149389 -> ((29 50) (8 53))
15: 165464 -> ((38 48) (20 54))
16: 171288 -> ((24 54) (17 55))
17: 195841 -> ((22 57) (9 58))
18: 216027 -> ((22 59) (3 60))
19: 216125 -> ((45 50) (5 60))
20: 262656 -> ((36 60) (8 64))
21: 314496 -> ((30 66) (4 68))
22: 320264 -> ((32 66) (18 68))
23: 327763 -> ((51 58) (30 67))
24: 373464 -> ((54 60) (6 72))
25: 402597 -> ((56 61) (42 69))
2000: 1671816384 -> ((940 944) (428 1168))
2001: 1672470592 -> ((632 1124) (29 1187))
2002: 1673170856 -> ((828 1034) (458 1164))
2003: 1675045225 -> ((744 1081) (522 1153))
2004: 1675958167 -> ((711 1096) (492 1159))
2005: 1676926719 -> ((714 1095) (63 1188))
2006: 1677646971 -> ((891 990) (99 1188))

Sidef

Translation of: Raku
var (start=1, end=25) = ARGV.map{.to_i}...
 
func display (h, start, end) {
    var i = start
    for n in [h.grep {|_,v| v.len > 1 }.keys.sort_by{.to_i}[start-1 .. end-1]] {
        printf("%4d %10d  =>\t%s\n", i++, n,
            h{n}.map{ "%4d³ + %-s" % (.first, "#{.last}³") }.join(",\t"))
    }
}
 
var taxi = Hash()
var taxis = 0
var terminate = 0
 
for c1 (1..Inf) {
    if (0<terminate && terminate<c1) {
        display(taxi, start, end)
        break
    }
    var c = c1**3
    for c2 (1..c1) {
        var this = (c2**3 + c)
        taxi{this} := [] << [c2, c1]
        ++taxis if (taxi{this}.len == 2)
        if (taxis==end && !terminate) {
            terminate = taxi.grep{|_,v| v.len > 1 }.keys.map{.to_i}.max.root(3)
        }
    }
}
Output:
   1       1729  =>	   9³ + 10³,	   1³ + 12³
   2       4104  =>	   9³ + 15³,	   2³ + 16³
   3      13832  =>	  18³ + 20³,	   2³ + 24³
   4      20683  =>	  19³ + 24³,	  10³ + 27³
   5      32832  =>	  18³ + 30³,	   4³ + 32³
   6      39312  =>	  15³ + 33³,	   2³ + 34³
   7      40033  =>	  16³ + 33³,	   9³ + 34³
   8      46683  =>	  27³ + 30³,	   3³ + 36³
   9      64232  =>	  26³ + 36³,	  17³ + 39³
  10      65728  =>	  31³ + 33³,	  12³ + 40³
  11     110656  =>	  36³ + 40³,	   4³ + 48³
  12     110808  =>	  27³ + 45³,	   6³ + 48³
  13     134379  =>	  38³ + 43³,	  12³ + 51³
  14     149389  =>	  29³ + 50³,	   8³ + 53³
  15     165464  =>	  38³ + 48³,	  20³ + 54³
  16     171288  =>	  24³ + 54³,	  17³ + 55³
  17     195841  =>	  22³ + 57³,	   9³ + 58³
  18     216027  =>	  22³ + 59³,	   3³ + 60³
  19     216125  =>	  45³ + 50³,	   5³ + 60³
  20     262656  =>	  36³ + 60³,	   8³ + 64³
  21     314496  =>	  30³ + 66³,	   4³ + 68³
  22     320264  =>	  32³ + 66³,	  18³ + 68³
  23     327763  =>	  51³ + 58³,	  30³ + 67³
  24     373464  =>	  54³ + 60³,	   6³ + 72³
  25     402597  =>	  56³ + 61³,	  42³ + 69³

With passed parameters 2000 and 2006:

2000 1671816384  =>	 940³ + 944³,	 428³ + 1168³
2001 1672470592  =>	 632³ + 1124³,	  29³ + 1187³
2002 1673170856  =>	 828³ + 1034³,	 458³ + 1164³
2003 1675045225  =>	 744³ + 1081³,	 522³ + 1153³
2004 1675958167  =>	 711³ + 1096³,	 492³ + 1159³
2005 1676926719  =>	 714³ + 1095³,	  63³ + 1188³
2006 1677646971  =>	 891³ + 990³,	  99³ + 1188³

Swift

extension Array {
  func combinations(_ k: Int) -> [[Element]] {
    return Self._combinations(slice: self[startIndex...], k)
  }

  static func _combinations(slice: Self.SubSequence, _ k: Int) -> [[Element]] {
    guard k != 1 else {
      return slice.map({ [$0] })
    }

    guard k != slice.count else {
      return [slice.map({ $0 })]
    }

    let chopped = slice[slice.index(after: slice.startIndex)...]

    var res = _combinations(slice: chopped, k - 1).map({ [[slice.first!], $0].flatMap({ $0 }) })

    res += _combinations(slice: chopped, k)

    return res
  }
}

let cubes = (1...).lazy.map({ $0 * $0 * $0 })
let taxis =
  Array(cubes.prefix(1201))
    .combinations(2)
    .reduce(into: [Int: [[Int]]](), { $0[$1[0] + $1[1], default: []].append($1) })


let res =
  taxis
    .lazy
    .filter({ $0.value.count > 1 })
    .sorted(by: { $0.key < $1.key })
    .map({ ($0.key, $0.value) })
    .prefix(2006)

print("First 25 taxicab numbers:")
for taxi in res[..<25] {
  print(taxi)
}

print("\n2000th through 2006th taxicab numbers:")
for taxi in res[1999..<2006] {
  print(taxi)
}
Output:
First 25 taxicab numbers:
(1729, [[1, 1728], [729, 1000]])
(4104, [[8, 4096], [729, 3375]])
(13832, [[8, 13824], [5832, 8000]])
(20683, [[1000, 19683], [6859, 13824]])
(32832, [[64, 32768], [5832, 27000]])
(39312, [[8, 39304], [3375, 35937]])
(40033, [[729, 39304], [4096, 35937]])
(46683, [[27, 46656], [19683, 27000]])
(64232, [[4913, 59319], [17576, 46656]])
(65728, [[1728, 64000], [29791, 35937]])
(110656, [[64, 110592], [46656, 64000]])
(110808, [[216, 110592], [19683, 91125]])
(134379, [[1728, 132651], [54872, 79507]])
(149389, [[512, 148877], [24389, 125000]])
(165464, [[8000, 157464], [54872, 110592]])
(171288, [[4913, 166375], [13824, 157464]])
(195841, [[729, 195112], [10648, 185193]])
(216027, [[27, 216000], [10648, 205379]])
(216125, [[125, 216000], [91125, 125000]])
(262656, [[512, 262144], [46656, 216000]])
(314496, [[64, 314432], [27000, 287496]])
(320264, [[5832, 314432], [32768, 287496]])
(327763, [[27000, 300763], [132651, 195112]])
(373464, [[216, 373248], [157464, 216000]])
(402597, [[74088, 328509], [175616, 226981]])

2000th through 2006th taxicab numbers:
(1671816384, [[78402752, 1593413632], [830584000, 841232384]])
(1672470592, [[24389, 1672446203], [252435968, 1420034624]])
(1673170856, [[96071912, 1577098944], [567663552, 1105507304]])
(1675045225, [[142236648, 1532808577], [411830784, 1263214441]])
(1675958167, [[119095488, 1556862679], [359425431, 1316532736]])
(1676926719, [[250047, 1676676672], [363994344, 1312932375]])
(1677646971, [[970299, 1676676672], [707347971, 970299000]])

Tcl

Works with: Tcl version 8.6
Translation of: Python
package require Tcl 8.6

proc heappush {heapName item} {
    upvar 1 $heapName heap
    set idx [lsearch -bisect -index 0 -integer $heap [lindex $item 0]]
    set heap [linsert $heap [expr {$idx + 1}] $item]
}
coroutine cubesum apply {{} {
    yield
    set h {}
    set n 1
    while true {
	while {![llength $h] || [lindex $h 0 0] > $n**3} {
	    heappush h [list [expr {$n**3 + 1}] $n 1]
	    incr n
	}
	set h [lassign $h item]
	yield $item
	lassign $item s x y
	if {[incr y] < $x} {
	    heappush h [list [expr {$x**3 + $y**3}] $x $y]
	}
    }
}}
coroutine taxis apply {{} {
    yield
    set out {{0 0 0}}
    while true {
	set s [cubesum]
	if {[lindex $s 0] == [lindex $out end 0]} {
	    lappend out $s
	} else {
	    if {[llength $out] > 1} {yield $out}
	    set out [list $s]
	}
    }
}}

# Put a cache in front for convenience
variable taxis {}
proc taxi {n} {
    variable taxis
    while {$n > [llength $taxis]} {lappend taxis [taxis]}
    return [lindex $taxis [expr {$n-1}]]
}

set 3 "\u00b3"
for {set n 1} {$n <= 25} {incr n} {
    puts ${n}:[join [lmap t [taxi $n] {format " %d = %d$3 + %d$3" {*}$t}] ","]
}
for {set n 2000} {$n <= 2006} {incr n} {
    puts ${n}:[join [lmap t [taxi $n] {format " %d = %d$3 + %d$3" {*}$t}] ","]
}
Output:
1: 1729 = 10³ + 9³, 1729 = 12³ + 1³
2: 4104 = 15³ + 9³, 4104 = 16³ + 2³
3: 13832 = 20³ + 18³, 13832 = 24³ + 2³
4: 20683 = 24³ + 19³, 20683 = 27³ + 10³
5: 32832 = 30³ + 18³, 32832 = 32³ + 4³
6: 39312 = 33³ + 15³, 39312 = 34³ + 2³
7: 40033 = 33³ + 16³, 40033 = 34³ + 9³
8: 46683 = 30³ + 27³, 46683 = 36³ + 3³
9: 64232 = 36³ + 26³, 64232 = 39³ + 17³
10: 65728 = 33³ + 31³, 65728 = 40³ + 12³
11: 110656 = 40³ + 36³, 110656 = 48³ + 4³
12: 110808 = 45³ + 27³, 110808 = 48³ + 6³
13: 134379 = 43³ + 38³, 134379 = 51³ + 12³
14: 149389 = 50³ + 29³, 149389 = 53³ + 8³
15: 165464 = 48³ + 38³, 165464 = 54³ + 20³
16: 171288 = 54³ + 24³, 171288 = 55³ + 17³
17: 195841 = 57³ + 22³, 195841 = 58³ + 9³
18: 216027 = 59³ + 22³, 216027 = 60³ + 3³
19: 216125 = 50³ + 45³, 216125 = 60³ + 5³
20: 262656 = 60³ + 36³, 262656 = 64³ + 8³
21: 314496 = 66³ + 30³, 314496 = 68³ + 4³
22: 320264 = 66³ + 32³, 320264 = 68³ + 18³
23: 327763 = 58³ + 51³, 327763 = 67³ + 30³
24: 373464 = 60³ + 54³, 373464 = 72³ + 6³
25: 402597 = 61³ + 56³, 402597 = 69³ + 42³
2000: 1671816384 = 944³ + 940³, 1671816384 = 1168³ + 428³
2001: 1672470592 = 1124³ + 632³, 1672470592 = 1187³ + 29³
2002: 1673170856 = 1034³ + 828³, 1673170856 = 1164³ + 458³
2003: 1675045225 = 1081³ + 744³, 1675045225 = 1153³ + 522³
2004: 1675958167 = 1096³ + 711³, 1675958167 = 1159³ + 492³
2005: 1676926719 = 1095³ + 714³, 1676926719 = 1188³ + 63³
2006: 1677646971 = 990³ + 891³, 1677646971 = 1188³ + 99³

VBA

Public Type tuple
    i As Variant
    j As Variant
    sum As Variant
End Type
Public Type tuple3
    i1 As Variant
    j1 As Variant
    i2 As Variant
    j2 As Variant
    i3 As Variant
    j3 As Variant
    sum As Variant
End Type
Sub taxicab_numbers()
    Dim i As Variant, j As Variant
    Dim k As Long
    Const MAX = 2019
    Dim p(MAX) As Variant
    Const bigMAX = (MAX + 1) * (MAX / 2)
    Dim big(1 To bigMAX) As tuple
    Const resMAX = 4400
    Dim res(1 To resMAX) As tuple3
    For i = 1 To MAX
        p(i) = CDec(i * i * i) 'convert Variant to Decimal
    Next i                     'wich hold numbers upto 10^28
    
    k = 1
    For i = 1 To MAX
        For j = i To MAX
            big(k).i = CDec(i)
            big(k).j = CDec(j)
            big(k).sum = CDec(p(i) + p(j))
            k = k + 1
        Next j
    Next i
    n = 1
    Quicksort big, LBound(big), UBound(big)
    For i = 1 To bigMAX - 1
        If big(i).sum = big(i + 1).sum Then
            res(n).i1 = CStr(big(i).i)
            res(n).j1 = CStr(big(i).j)
            res(n).i2 = CStr(big(i + 1).i)
            res(n).j2 = CStr(big(i + 1).j)
            If big(i + 1).sum = big(i + 2).sum Then
                res(n).i3 = CStr(big(i + 2).i)
                res(n).j3 = CStr(big(i + 2).j)
                i = i + 1
            End If
            res(n).sum = CStr(big(i).sum)
            n = n + 1
            i = i + 1
        End If
    Next i
    Debug.Print n - 1; " taxis"
    For i = 1 To 25
        With res(i)
            Debug.Print String$(4 - Len(CStr(i)), " "); i;
            Debug.Print String$(11 - Len(.sum), " "); .sum; " = ";
            Debug.Print String$(4 - Len(.i1), " "); .i1; "^3 +";
            Debug.Print String$(4 - Len(.j1), " "); .j1; "^3 = ";
            Debug.Print String$(4 - Len(.i2), " "); .i2; "^3 +";
            Debug.Print String$(4 - Len(.j2), " "); .j2; "^3"
        End With
    Next i
    Debug.Print
    For i = 2000 To 2006
        With res(i)
            Debug.Print String$(4 - Len(CStr(i)), " "); i;
            Debug.Print String$(11 - Len(.sum), " "); .sum; " = ";
            Debug.Print String$(4 - Len(.i1), " "); .i1; "^3 +";
            Debug.Print String$(4 - Len(.j1), " "); .j1; "^3 = ";
            Debug.Print String$(4 - Len(.i2), " "); .i2; "^3 +";
            Debug.Print String$(4 - Len(.j2), " "); .j2; "^3"
        End With

    Next i
    Debug.Print
    For i = 1 To resMAX
        If res(i).i3 <> "" Then
            With res(i)
                Debug.Print String$(4 - Len(CStr(i)), " "); i;
                Debug.Print String$(11 - Len(.sum), " "); .sum; " = ";
                Debug.Print String$(4 - Len(.i1), " "); .i1; "^3 +";
                Debug.Print String$(4 - Len(.j1), " "); .j1; "^3 = ";
                Debug.Print String$(4 - Len(.i2), " "); .i2; "^3 +";
                Debug.Print String$(4 - Len(.j2), " "); .j2; "^3";
                Debug.Print String$(4 - Len(.i3), " "); .i3; "^3 +";
                Debug.Print String$(4 - Len(.j3), " "); .j3; "^3"
            End With
        End If
    Next i
End Sub
Sub Quicksort(vArray() As tuple, arrLbound As Long, arrUbound As Long)
    'https://wellsr.com/vba/2018/excel/vba-quicksort-macro-to-sort-arrays-fast/
    'Sorts a one-dimensional VBA array from smallest to largest
    'using a very fast quicksort algorithm variant.
    'Adapted to multidimensions/typedef
    Dim pivotVal As Variant
    Dim vSwap    As tuple
    Dim tmpLow   As Long
    Dim tmpHi    As Long
     
    tmpLow = arrLbound
    tmpHi = arrUbound
    pivotVal = vArray((arrLbound + arrUbound) \ 2).sum
     
    While (tmpLow <= tmpHi) 'divide
        While (vArray(tmpLow).sum < pivotVal And tmpLow < arrUbound)
            tmpLow = tmpLow + 1
        Wend
        
        While (pivotVal < vArray(tmpHi).sum And tmpHi > arrLbound)
            tmpHi = tmpHi - 1
        Wend
     
        If (tmpLow <= tmpHi) Then
             vSwap.i = vArray(tmpLow).i
             vSwap.j = vArray(tmpLow).j
             vSwap.sum = vArray(tmpLow).sum
             vArray(tmpLow).i = vArray(tmpHi).i
             vArray(tmpLow).j = vArray(tmpHi).j
             vArray(tmpLow).sum = vArray(tmpHi).sum
             vArray(tmpHi).i = vSwap.i
             vArray(tmpHi).j = vSwap.j
             vArray(tmpHi).sum = vSwap.sum
             tmpLow = tmpLow + 1
             tmpHi = tmpHi - 1
        End If
    Wend
 
    If (arrLbound < tmpHi) Then Quicksort vArray, arrLbound, tmpHi 'conquer
    If (tmpLow < arrUbound) Then Quicksort vArray, tmpLow, arrUbound 'conquer
End Sub
Output:
 4399  taxis
    1        1729 =    9^3 +  10^3 =    1^3 +  12^3
    2        4104 =    2^3 +  16^3 =    9^3 +  15^3
    3       13832 =    2^3 +  24^3 =   18^3 +  20^3
    4       20683 =   19^3 +  24^3 =   10^3 +  27^3
    5       32832 =   18^3 +  30^3 =    4^3 +  32^3
    6       39312 =   15^3 +  33^3 =    2^3 +  34^3
    7       40033 =   16^3 +  33^3 =    9^3 +  34^3
    8       46683 =   27^3 +  30^3 =    3^3 +  36^3
    9       64232 =   26^3 +  36^3 =   17^3 +  39^3
   10       65728 =   31^3 +  33^3 =   12^3 +  40^3
   11      110656 =    4^3 +  48^3 =   36^3 +  40^3
   12      110808 =   27^3 +  45^3 =    6^3 +  48^3
   13      134379 =   12^3 +  51^3 =   38^3 +  43^3
   14      149389 =   29^3 +  50^3 =    8^3 +  53^3
   15      165464 =   38^3 +  48^3 =   20^3 +  54^3
   16      171288 =   24^3 +  54^3 =   17^3 +  55^3
   17      195841 =    9^3 +  58^3 =   22^3 +  57^3
   18      216027 =   22^3 +  59^3 =    3^3 +  60^3
   19      216125 =   45^3 +  50^3 =    5^3 +  60^3
   20      262656 =   36^3 +  60^3 =    8^3 +  64^3
   21      314496 =    4^3 +  68^3 =   30^3 +  66^3
   22      320264 =   32^3 +  66^3 =   18^3 +  68^3
   23      327763 =   51^3 +  58^3 =   30^3 +  67^3
   24      373464 =   54^3 +  60^3 =    6^3 +  72^3
   25      402597 =   56^3 +  61^3 =   42^3 +  69^3

 2000  1671816384 =  940^3 + 944^3 =  428^3 +1168^3
 2001  1672470592 =   29^3 +1187^3 =  632^3 +1124^3
 2002  1673170856 =  828^3 +1034^3 =  458^3 +1164^3
 2003  1675045225 =  744^3 +1081^3 =  522^3 +1153^3
 2004  1675958167 =  492^3 +1159^3 =  711^3 +1096^3
 2005  1676926719 =  714^3 +1095^3 =   63^3 +1188^3
 2006  1677646971 =   99^3 +1188^3 =  891^3 + 990^3

  455    87539319 =  167^3 + 436^3 =  228^3 + 423^3 255^3 + 414^3
  535   119824488 =   90^3 + 492^3 =  346^3 + 428^3  11^3 + 493^3
  588   143604279 =  408^3 + 423^3 =  359^3 + 460^3 111^3 + 522^3
  655   175959000 =   70^3 + 560^3 =  315^3 + 525^3 198^3 + 552^3
  888   327763000 =  300^3 + 670^3 =  339^3 + 661^3 510^3 + 580^3
 1299   700314552 =  334^3 + 872^3 =  456^3 + 846^3 510^3 + 828^3
 1398   804360375 =   15^3 + 930^3 =  295^3 + 920^3 198^3 + 927^3
 1515   958595904 =   22^3 + 986^3 =  180^3 + 984^3 692^3 + 856^3
 1660  1148834232 =  718^3 + 920^3 =  816^3 + 846^3 222^3 +1044^3
 1837  1407672000 =  140^3 +1120^3 =  396^3 +1104^3 630^3 +1050^3
 2100  1840667192 =  681^3 +1151^3 =  372^3 +1214^3 225^3 +1223^3
 2143  1915865217 =    9^3 +1242^3 =  484^3 +1217^3 969^3 +1002^3
 2365  2363561613 =  501^3 +1308^3 =  684^3 +1269^3 765^3 +1242^3
 2480  2622104000 = 1020^3 +1160^3 =  600^3 +1340^3 678^3 +1322^3
 2670  3080802816 =  904^3 +1328^3 =   81^3 +1455^3 456^3 +1440^3
 2732  3235261176 =   33^3 +1479^3 =  270^3 +1476^31038^3 +1284^3
 2845  3499524728 =  116^3 +1518^3 =  350^3 +1512^31169^3 +1239^3
 2895  3623721192 =  348^3 +1530^3 =  761^3 +1471^31098^3 +1320^3
 2979  3877315533 = 1224^3 +1269^3 = 1077^3 +1380^3 333^3 +1566^3
 3293  4750893000 =  210^3 +1680^3 =  945^3 +1575^3 594^3 +1656^3
 3562  5544709352 =  207^3 +1769^3 = 1076^3 +1626^3 842^3 +1704^3
 3589  5602516416 =  912^3 +1692^3 = 1020^3 +1656^3 668^3 +1744^3
 3826  6434883000 =  590^3 +1840^3 =   30^3 +1860^3 396^3 +1854^3
 4162  7668767232 =   44^3 +1972^3 = 1384^3 +1712^3 360^3 +1968^3
 4359  8849601000 = 1017^3 +1983^3 = 1530^3 +1740^3 900^3 +2010^3

Visual Basic .NET

Translation of: C#
Imports System.Text

Module Module1

    Function GetTaxicabNumbers(length As Integer) As IDictionary(Of Long, IList(Of Tuple(Of Integer, Integer)))
        Dim sumsOfTwoCubes As New SortedList(Of Long, IList(Of Tuple(Of Integer, Integer)))

        For i = 1 To Integer.MaxValue - 1
            For j = 1 To Integer.MaxValue - 1
                Dim sum = CLng(Math.Pow(i, 3) + Math.Pow(j, 3))

                If Not sumsOfTwoCubes.ContainsKey(sum) Then
                    sumsOfTwoCubes.Add(sum, New List(Of Tuple(Of Integer, Integer)))
                End If

                sumsOfTwoCubes(sum).Add(Tuple.Create(i, j))

                If j >= i Then
                    Exit For
                End If
            Next

            REM Found that you need to keep going for a while after the length, because higher i values fill in gaps
            If sumsOfTwoCubes.AsEnumerable.Count(Function(t) t.Value.Count >= 2) >= length * 1.1 Then
                Exit For
            End If
        Next

        Dim values = (From t In sumsOfTwoCubes Where t.Value.Count >= 2 Select t) _
            .Take(2006) _
            .ToDictionary(Function(u) u.Key, Function(u) u.Value)
        Return values
    End Function

    Sub PrintTaxicabNumbers(values As IDictionary(Of Long, IList(Of Tuple(Of Integer, Integer))))
        Dim i = 1
        For Each taxicabNumber In values.Keys
            Dim output As New StringBuilder
            output.AppendFormat("{0,10}" + vbTab + "{1,4}", i, taxicabNumber)

            For Each numbers In values(taxicabNumber)
                output.AppendFormat(vbTab + "= {0}^3 + {1}^3", numbers.Item1, numbers.Item2)
            Next

            If i <= 25 OrElse (i >= 2000 AndAlso i <= 2006) Then
                Console.WriteLine(output.ToString)
            End If

            i += 1
        Next
    End Sub

    Sub Main()
        Dim taxicabNumbers = GetTaxicabNumbers(2006)
        PrintTaxicabNumbers(taxicabNumbers)
    End Sub

End Module
Output:
         1      1729    = 10^3 + 9^3    = 12^3 + 1^3
         2      4104    = 15^3 + 9^3    = 16^3 + 2^3
         3      13832   = 20^3 + 18^3   = 24^3 + 2^3
         4      20683   = 24^3 + 19^3   = 27^3 + 10^3
         5      32832   = 30^3 + 18^3   = 32^3 + 4^3
         6      39312   = 33^3 + 15^3   = 34^3 + 2^3
         7      40033   = 33^3 + 16^3   = 34^3 + 9^3
         8      46683   = 30^3 + 27^3   = 36^3 + 3^3
         9      64232   = 36^3 + 26^3   = 39^3 + 17^3
        10      65728   = 33^3 + 31^3   = 40^3 + 12^3
        11      110656  = 40^3 + 36^3   = 48^3 + 4^3
        12      110808  = 45^3 + 27^3   = 48^3 + 6^3
        13      134379  = 43^3 + 38^3   = 51^3 + 12^3
        14      149389  = 50^3 + 29^3   = 53^3 + 8^3
        15      165464  = 48^3 + 38^3   = 54^3 + 20^3
        16      171288  = 54^3 + 24^3   = 55^3 + 17^3
        17      195841  = 57^3 + 22^3   = 58^3 + 9^3
        18      216027  = 59^3 + 22^3   = 60^3 + 3^3
        19      216125  = 50^3 + 45^3   = 60^3 + 5^3
        20      262656  = 60^3 + 36^3   = 64^3 + 8^3
        21      314496  = 66^3 + 30^3   = 68^3 + 4^3
        22      320264  = 66^3 + 32^3   = 68^3 + 18^3
        23      327763  = 58^3 + 51^3   = 67^3 + 30^3
        24      373464  = 60^3 + 54^3   = 72^3 + 6^3
        25      402597  = 61^3 + 56^3   = 69^3 + 42^3
      2000      1671816384      = 944^3 + 940^3 = 1168^3 + 428^3
      2001      1672470592      = 1124^3 + 632^3        = 1187^3 + 29^3
      2002      1673170856      = 1034^3 + 828^3        = 1164^3 + 458^3
      2003      1675045225      = 1081^3 + 744^3        = 1153^3 + 522^3
      2004      1675958167      = 1096^3 + 711^3        = 1159^3 + 492^3
      2005      1676926719      = 1095^3 + 714^3        = 1188^3 + 63^3
      2006      1677646971      = 990^3 + 891^3 = 1188^3 + 99^3

Wren

Library: Wren-sort
Library: Wren-fmt
import "./sort" for Sort
import "./fmt" for Fmt

var cubesSum = {}
var taxicabs = []

for (i in 1..1199) {
    for (j in i+1..1200) {
        var sum = i*i*i + j*j*j
        if (!cubesSum[sum]) {
            cubesSum[sum] = [i, j]
        } else {
            taxicabs.add([sum, cubesSum[sum], [i, j]])
        }
    }
}
var cmp = Fn.new { |a, b| (a[0] - b[0]).sign }
Sort.quick(taxicabs, 0, taxicabs.count-1, cmp)
// remove those numbers which have additional pairs of cubes
for (i in taxicabs.count-2..0) {
    if (taxicabs[i][0] == taxicabs[i+1][0]) taxicabs.removeAt(i+1)
}

System.print("The first 25 taxicab numbers are:")
for (i in 1..25) {
    var t = taxicabs[i-1]
    Fmt.lprint("$2d: $,7d = $2d³ + $2d³ = $2d³ + $2d³", [i, t[0], t[1][0], t[1][1], t[2][0], t[2][1]])
}

System.print("\nThe 2,000th to 2,006th taxicab numbers are:")
for (i in 2000..2006) {
    var t = taxicabs[i-1]
    Fmt.lprint("$,5d: $,13d = $3d³ + $,5d³ = $3d³ + $,5d³", [i, t[0], t[1][0], t[1][1], t[2][0], t[2][1]])
}
Output:
The first 25 taxicab numbers are:
 1:   1,729 =  1³ + 12³ =  9³ + 10³
 2:   4,104 =  2³ + 16³ =  9³ + 15³
 3:  13,832 =  2³ + 24³ = 18³ + 20³
 4:  20,683 = 10³ + 27³ = 19³ + 24³
 5:  32,832 =  4³ + 32³ = 18³ + 30³
 6:  39,312 =  2³ + 34³ = 15³ + 33³
 7:  40,033 =  9³ + 34³ = 16³ + 33³
 8:  46,683 =  3³ + 36³ = 27³ + 30³
 9:  64,232 = 17³ + 39³ = 26³ + 36³
10:  65,728 = 12³ + 40³ = 31³ + 33³
11: 110,656 =  4³ + 48³ = 36³ + 40³
12: 110,808 =  6³ + 48³ = 27³ + 45³
13: 134,379 = 12³ + 51³ = 38³ + 43³
14: 149,389 =  8³ + 53³ = 29³ + 50³
15: 165,464 = 20³ + 54³ = 38³ + 48³
16: 171,288 = 17³ + 55³ = 24³ + 54³
17: 195,841 =  9³ + 58³ = 22³ + 57³
18: 216,027 =  3³ + 60³ = 22³ + 59³
19: 216,125 =  5³ + 60³ = 45³ + 50³
20: 262,656 =  8³ + 64³ = 36³ + 60³
21: 314,496 =  4³ + 68³ = 30³ + 66³
22: 320,264 = 18³ + 68³ = 32³ + 66³
23: 327,763 = 30³ + 67³ = 51³ + 58³
24: 373,464 =  6³ + 72³ = 54³ + 60³
25: 402,597 = 42³ + 69³ = 56³ + 61³

The 2,000th to 2,006th taxicab numbers are:
2,000: 1,671,816,384 = 428³ + 1,168³ = 940³ +   944³
2,001: 1,672,470,592 =  29³ + 1,187³ = 632³ + 1,124³
2,002: 1,673,170,856 = 458³ + 1,164³ = 828³ + 1,034³
2,003: 1,675,045,225 = 522³ + 1,153³ = 744³ + 1,081³
2,004: 1,675,958,167 = 492³ + 1,159³ = 711³ + 1,096³
2,005: 1,676,926,719 =  63³ + 1,188³ = 714³ + 1,095³
2,006: 1,677,646,971 =  99³ + 1,188³ = 891³ +   990³

XPL0

Slow, brute force solution.

int N, I, J, SI, SJ, Count, Tally;
[Count:= 0;  N:= 0;
repeat  Tally:= 0;
        I:= 1;
        repeat  J:= I+1;
                repeat  if N = I*I*I + J*J*J then
                            [Tally:= Tally+1;
                            if Tally >= 2 then
                                [Count:= Count+1;
                                IntOut(0, Count);  Text(0, ": ");
                                IntOut(0, N);  Text(0, " = ");
                                IntOut(0, SI); Text(0, "^^3 + ");
                                IntOut(0, SJ); Text(0, "^^3 = ");
                                IntOut(0, I);  Text(0, "^^3 + ");
                                IntOut(0, J);  Text(0, "^^3");
                                CrLf(0);
                                J:= 1000;  I:= J;
                                ];
                            SI:= I;  SJ:= J;
                            ];
                        J:= J+1;
                until   I*I*I + J*J*J > N;
                I:= I+1;
        until   I*I*I*2 > N;
        N:= N+1;
until   Count >= 25;
]
Output:
1: 1729 = 1^3 + 12^3 = 9^3 + 10^3
2: 4104 = 2^3 + 16^3 = 9^3 + 15^3
3: 13832 = 2^3 + 24^3 = 18^3 + 20^3
4: 20683 = 10^3 + 27^3 = 19^3 + 24^3
5: 32832 = 4^3 + 32^3 = 18^3 + 30^3
6: 39312 = 2^3 + 34^3 = 15^3 + 33^3
7: 40033 = 9^3 + 34^3 = 16^3 + 33^3
8: 46683 = 3^3 + 36^3 = 27^3 + 30^3
9: 64232 = 17^3 + 39^3 = 26^3 + 36^3
10: 65728 = 12^3 + 40^3 = 31^3 + 33^3
11: 110656 = 4^3 + 48^3 = 36^3 + 40^3
12: 110808 = 6^3 + 48^3 = 27^3 + 45^3
13: 134379 = 12^3 + 51^3 = 38^3 + 43^3
14: 149389 = 8^3 + 53^3 = 29^3 + 50^3
15: 165464 = 20^3 + 54^3 = 38^3 + 48^3
16: 171288 = 17^3 + 55^3 = 24^3 + 54^3
17: 195841 = 9^3 + 58^3 = 22^3 + 57^3
18: 216027 = 3^3 + 60^3 = 22^3 + 59^3
19: 216125 = 5^3 + 60^3 = 45^3 + 50^3
20: 262656 = 8^3 + 64^3 = 36^3 + 60^3
21: 314496 = 4^3 + 68^3 = 30^3 + 66^3
22: 320264 = 18^3 + 68^3 = 32^3 + 66^3
23: 327763 = 30^3 + 67^3 = 51^3 + 58^3
24: 373464 = 6^3 + 72^3 = 54^3 + 60^3
25: 402597 = 42^3 + 69^3 = 56^3 + 61^3

zkl

Translation of: D

An array of bytes is used to hold n, where array[n³+m³]==n.

fcn taxiCabNumbers{
   const HeapSZ=0d5_000_000;
   iCubes:=[1..120].apply("pow",3);
   sum2cubes:=Data(HeapSZ).fill(0);  // BFheap of 1 byte zeros
   taxiNums:=List();
   foreach i,i3 in ([1..].zip(iCubes)){
      foreach j,j3 in ([i+1..].zip(iCubes[i,*])){
         ij3:=i3+j3; 
	 if(z:=sum2cubes[ij3]){
	    taxiNums.append(T(ij3,
		z,(ij3-z.pow(3)).toFloat().pow(1.0/3).round().toInt(),
		i,j));
	 }
	 else sum2cubes[ij3]=i;
      }
   }
   taxiNums.sort(fcn([(a,_)],[(b,_)]){ a<b })
}
fcn print(n,taxiNums){
   [n..].zip(taxiNums).pump(Console.println,fcn([(n,t)]){ 
      "%4d: %10,d = %2d\u00b3 + %2d\u00b3 =  %2d\u00b3 + %2d\u00b3".fmt(n,t.xplode())
   })
}
taxiNums:=taxiCabNumbers();  // 63 pairs
taxiNums[0,25]:print(1,_);
Output:
   1:      1,729 =  1³ + 12³ =   9³ + 10³
   2:      4,104 =  2³ + 16³ =   9³ + 15³
   3:     13,832 =  2³ + 24³ =  18³ + 20³
   4:     20,683 = 10³ + 27³ =  19³ + 24³
   5:     32,832 =  4³ + 32³ =  18³ + 30³
   6:     39,312 =  2³ + 34³ =  15³ + 33³
   7:     40,033 =  9³ + 34³ =  16³ + 33³
   8:     46,683 =  3³ + 36³ =  27³ + 30³
   9:     64,232 = 17³ + 39³ =  26³ + 36³
  10:     65,728 = 12³ + 40³ =  31³ + 33³
  11:    110,656 =  4³ + 48³ =  36³ + 40³
  12:    110,808 =  6³ + 48³ =  27³ + 45³
  13:    134,379 = 12³ + 51³ =  38³ + 43³
  14:    149,389 =  8³ + 53³ =  29³ + 50³
  15:    165,464 = 20³ + 54³ =  38³ + 48³
  16:    171,288 = 17³ + 55³ =  24³ + 54³
  17:    195,841 =  9³ + 58³ =  22³ + 57³
  18:    216,027 =  3³ + 60³ =  22³ + 59³
  19:    216,125 =  5³ + 60³ =  45³ + 50³
  20:    262,656 =  8³ + 64³ =  36³ + 60³
  21:    314,496 =  4³ + 68³ =  30³ + 66³
  22:    320,264 = 18³ + 68³ =  32³ + 66³
  23:    327,763 = 30³ + 67³ =  51³ + 58³
  24:    373,464 =  6³ + 72³ =  54³ + 60³
  25:    402,597 = 42³ + 69³ =  56³ + 61³
Translation of: Python

Using a binary heap:

fcn cubeSum{
   heap,n:=Heap(fcn([(a,_)],[(b,_)]){ a<=b }), 1;  // heap cnt maxes out @ 244
   while(1){
      while(heap.empty or heap.top[0]>n.pow(3)){ # could also pre-calculate cubes
	 heap.push(T(n.pow(3) + 1, n,1));
	 n+=1;
      }
      s,x,y:= sxy:=heap.pop();
      vm.yield(sxy);
      y+=1;
      if(y<x)    # should be y <= x?
	 heap.push(T(x.pow(3) + y.pow(3), x,y));
   }
}
fcn taxis{
   out:=List(T(0,0,0));
   foreach s in (Utils.Generator(cubeSum)){
      if(s[0]==out[-1][0]) out.append(s);
      else{
	 if(out.len()>1) vm.yield(out);
	 out.clear(s)
      }
   }
} 
n:=0;
foreach x in (Utils.Generator(taxis)){
   n += 1;
   if(n >= 2006) break;
   if(n <= 25 or n >= 2000) println(n,": ",x);
}

And a quickie heap implementation:

class Heap{  // binary heap
   fcn init(lteqFcn='<=){
      var [const, private] heap=List().pad(64,Void); // a power of 2
      var cnt=0, cmp=lteqFcn;
   }
   fcn push(v){
	// Resize the heap if it is too small to hold another item
      if (cnt==heap.len()) heap.pad(cnt*2,Void);

      index:=cnt; cnt+=1; while(index){	 // Find out where to put the element
	 parent:=(index - 1)/2;
	 if(cmp(heap[parent],v)) break;
	 heap[index] = heap[parent];
	 index = parent;
      }
      heap[index] = v;
   }
   fcn pop{  // Remove the biggest element and return it
      if(not cnt) return(Void);
      v,temp:=heap[0], heap[cnt-=1];

      // Reorder the elements
      index:=0; while(1){   // Find the child to swap with
	 swap:=index*2 + 1;
	 if (swap>=cnt) break; // If there are no children, the heap is reordered
	 other:=swap + 1;
	 if(other<cnt and cmp(heap[other],heap[swap])) swap = other;
	 if(cmp(temp,heap[swap])) break; // If the bigger child is less than or equal to its parent, the heap is reordered

	 heap[index]=heap[swap];
	 index = swap;
      }
      heap[index] = temp;
      v
   }
   var [proxy] top=fcn  { if(cnt==0) Void else heap[0] };
   var [proxy] empty=fcn{ (not cnt) };
}
Output:
1: L(L(1729,10,9),L(1729,12,1))
...
23: L(L(327763,67,30),L(327763,58,51))
24: L(L(373464,60,54),L(373464,72,6))
25: L(L(402597,61,56),L(402597,69,42))
2000: L(L(1671816384,944,940),L(1671816384,1168,428))
2001: L(L(1672470592,1124,632),L(1672470592,1187,29))
2002: L(L(1673170856,1034,828),L(1673170856,1164,458))
2003: L(L(1675045225,1153,522),L(1675045225,1081,744))
2004: L(L(1675958167,1096,711),L(1675958167,1159,492))
2005: L(L(1676926719,1188,63),L(1676926719,1095,714))

ZX Spectrum Basic

This will, in the strictest sense, work. Don't hold your breath though; after six hours on an emulator at full speed it had generated the first 10 numbers. Getting to 2006 may take a while longer.

You cannot fit the whole 1625-entry table of cubes (and this program on top) into the 16K ZX Spectrum. Replace all 1625s with 1200s to resolve; numerically unjustified as an exhaustive search, but we know this will be sufficient to find the 2006th number. Eventually.

10 DIM f(1625): REM populating a cube table at the start will be faster than computing the cubes on the fly
20 FOR x=1 TO 1625
30 LET f(x)=x*x*x: REM x*x*x rather than x^3 as the ZX Spectrum's exponentiation function is legendarily slow
40 NEXT x
50 LET c=0
60 FOR x=1 TO 4294967295: REM the highest number the ZX Spectrum Basic can accurately hold internally; floor (cuberoot max)=1625, hence the table limit
70 LET k=0
80 FOR m=1 TO 1625
90 FOR n=m+1 TO 1625
100 IF f(m)+f(n)=x THEN GOTO 160
110 IF f(n)>=x THEN LET n=1625: REM overshot, break out of the loop
120 IF f(m)>=x THEN LET m=1625
130 NEXT n
140 NEXT m
150 NEXT x
160 IF k=1 THEN LET q=m: LET r=n: GO TO 230: REM got one!
170 LET o=m
180 LET p=n
190 LET k=1
200 NEXT n
210 NEXT m
220 NEXT x
230 LET c=c+1
240 IF c>25 AND c<2000 THEN GO TO 330
250 LET t$="": REM convert number to string; while ZX Spectrum Basic can store all the digits of integers up to 2^32-1...
260 LET t=INT (x/100000): REM ...it will resort to scientific notation trying to display any more than eight digits
270 LET b=x-t*100000
280 IF t=0 THEN GO TO 300: REM omit leading zero
290 LET t$=STR$ t
300 LET t$=t$+STR$ b
310 PRINT c;":";t$;"=";q;"^3+";r;"^3=";o;"^3+";p;"^3"
320 POKE 23692,10: REM suppress "scroll?" prompt when screen fills up at c=22
330 IF c=2006 THEN LET x=4294967295: LET n=1625: LET m=1625
340 NEXT n
350 NEXT m
360 NEXT x
Output:
1:1729=9^3+10^3=1^3+12^3
2:4104=9^3+15^3=2^3+16^3
3:13832=18^3+20^3=2^3+24^3
4:20683=19^3+24^3=10^3+27^3
5:32832=18^3+30^3=4^3+32^3
6:39312=15^3+33^3=2^3+34^3
7:40033=16^3+33^3=9^3+34^3
8:46683=27^3+30^3=3^3+36^3
9:64232=26^3+36^3=17^3+39^3
10:65728=31^3+33^3=12^3+40^3

D BREAK into program, 100:1

This program produces the first 25 Taxicab numbers. It is written with speed in mind. The runtime is about 45 minutes on a ZX Spectrum (3.5 Mhz).

  10 LET T=0: DIM F(72): LET D=0: LET S=0: LET B=0: LET A=0: LET C=0
  20 DIM H(50): DIM Y(50,2): FOR D=1 TO 72: LET F(D)=D*D*D: NEXT D
  30 FOR A=1 TO 58: FOR B=A+1 TO 72: LET S=F(A)+F(B): FOR D=B-1 TO A STEP -1
  40 LET T=S-F(D): IF T>F(D) THEN NEXT B: NEXT A: GO TO 90
  45 IF s>405224 THEN GO TO 70
  50 IF F(INT (EXP (LN (T)/3)+.5))=T THEN GO TO 80
  60 NEXT D
  70 NEXT B: NEXT A: GO TO 90
  80 PRINT S,: LET C=C+1: LET H(C)=S: LET Y(C,1)=A*65536+B: LET Y(C,2)=INT (EXP (LN (T)/3)+.5)*65536+D: GO TO 70
  90 LET S=INT (C/2)
 100 LET T=0: FOR A=1 TO C-S: IF H(A)>H(A+S) THEN LET T=H(A): LET H(A)=H(A+S): LET H(A+S)=T: LET T=Y(A,1): LET Y(A,1)=Y(A+S,1): LET Y(A+S,1)=T: LET T=Y(A,2): LET Y(A,2)=Y(A+S,2): LET Y(A+S,2)=T
 110 NEXT A: IF T<>0 THEN GO TO 100
 120 IF S<>1 THEN LET S=INT (S/2): GO TO 100
 130 CLS : FOR A=1 TO 25: PRINT A;":";H(A);"=";
 131 LPRINT A;":";H(A);"=";:
 140 LET T=INT (Y(A,1)/65536): PRINT T;"^3+";Y(A,1)-T*65536;"^3=";
 141 LPRINT T;"^3+";Y(A,1)-T*65536;"^3=";
 150 LET T=INT (Y(A,2)/65536): PRINT T;"^3+";Y(A,2)-T*65536;"^3"
 151 LPRINT T;"^3+";Y(A,2)-T*65536;"^3"
 160 NEXT A: PRINT
 170 STOP
Output:
1:1729=1^3+12^3=9^3+10^3
2:4104=2^3+16^3=9^3+15^3
3:13832=2^3+24^3=18^3+20^3
4:20683=10^3+27^3=19^3+24^3
5:32832=4^3+32^3=18^3+30^3
6:39312=2^3+34^3=15^3+33^3
7:40033=9^3+34^3=16^3+33^3
8:46683=3^3+36^3=27^3+30^3
9:64232=17^3+39^3=26^3+36^3
10:65728=12^3+40^3=31^3+33^3
11:110656=4^3+48^3=36^3+40^3
12:110808=6^3+48^3=27^3+45^3
13:134379=12^3+51^3=38^3+43^3
14:149389=8^3+53^3=29^3+50^3
15:165464=20^3+54^3=38^3+48^3
16:171288=17^3+55^3=24^3+54^3
17:195841=9^3+58^3=22^3+57^3
18:216027=3^3+60^3=22^3+59^3
19:216125=5^3+60^3=45^3+50^3
20:262656=8^3+64^3=36^3+60^3
21:314496=4^3+68^3=30^3+66^3
22:320264=18^3+68^3=32^3+66^3
23:327763=30^3+67^3=51^3+58^3
24:373464=6^3+72^3=54^3+60^3
25:402597=42^3+69^3=56^3+61^3