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# Hamming numbers

Hamming numbers
You are encouraged to solve this task according to the task description, using any language you may know.

Hamming numbers are numbers of the form

$H = 2^i \cdot 3^j \cdot 5^k, \; \mathrm{where} \; i, j, k \geq 0$.

Hamming numbers are also known as ugly numbers and also 5-smooth numbers   (numbers whose prime divisors are less or equal to 5).

Generate the sequence of Hamming numbers, in increasing order. In particular:

1. Show the first twenty Hamming numbers.
2. Show the 1691st Hamming number (the last one below 231).
3. Show the one millionth Hamming number (if the language – or a convenient library – supports arbitrary-precision integers).

References

1. wp:Hamming_numbers
2. wp:Smooth_number
3. Hamming problem from Dr. Dobb's CodeTalk (dead link as of Sep 2011; parts of the thread here and here).

## Contents

Works with: GNAT

GNAT provides the datatypes Integer, Long_Integer and Long_Long_Integer.

Values for GNAT Pro 6.3.1, 64 bit Linux version:

• Integer covers the range -2**31 .. 2**31-1 (-2147483648 .. 2147483647).
• Long_Integer and Long_Long_Integer cover the range -2**63 .. 2**63-1 (-9223372036854775808 .. 9223372036854775807).

Using your own modular integer type (for example type My_Unsigned_Integer is mod 2**64;), you can expand the range to 0 .. 18446744073709551615, but this still is not enough for the millionth Hamming number.

For bigger numbers, you have to use an external library, for example Big_Number.

The code for calculating the Hamming numbers is kept generic, to easily expand the range by changing the concrete type.

with Ada.Text_IO;procedure Hamming is   generic      type Int_Type is private;      Zero  : Int_Type;      One   : Int_Type;      Two   : Int_Type;      Three : Int_Type;      Five  : Int_Type;      with function "mod" (Left, Right : Int_Type) return Int_Type is <>;      with function "/"   (Left, Right : Int_Type) return Int_Type is <>;      with function "+"   (Left, Right : Int_Type) return Int_Type is <>;   function Get_Hamming (Position : Positive) return Int_Type;    function Get_Hamming (Position : Positive) return Int_Type is      function Is_Hamming (Number : Int_Type) return Boolean is         Temporary : Int_Type := Number;      begin         while Temporary mod Two = Zero loop            Temporary := Temporary / Two;         end loop;         while Temporary mod Three = Zero loop            Temporary := Temporary / Three;         end loop;         while Temporary mod Five = Zero loop            Temporary := Temporary / Five;         end loop;         return Temporary = One;      end Is_Hamming;      Result   : Int_Type := One;      Previous : Positive := 1;   begin      while Previous /= Position loop         Result := Result + One;         if Is_Hamming (Result) then            Previous := Previous + 1;         end if;      end loop;      return Result;   end Get_Hamming;    -- up to 2**32 - 1   function Integer_Get_Hamming is new Get_Hamming      (Int_Type => Integer,       Zero     => 0,       One      => 1,       Two      => 2,       Three    => 3,       Five     => 5);    -- up to 2**64 - 1   function Long_Long_Integer_Get_Hamming is new Get_Hamming      (Int_Type => Long_Long_Integer,       Zero     => 0,       One      => 1,       Two      => 2,       Three    => 3,       Five     => 5);begin   Ada.Text_IO.Put ("1) First 20 Hamming numbers: ");   for I in 1 .. 20 loop      Ada.Text_IO.Put (Integer'Image (Integer_Get_Hamming (I)));   end loop;   Ada.Text_IO.New_Line;   Ada.Text_IO.Put_Line ("2) 1_691st Hamming number: " &      Integer'Image (Integer_Get_Hamming (1_691)));   -- even Long_Long_Integer overflows here   Ada.Text_IO.Put_Line ("3) 1_000_000st Hamming number: " &      Long_Long_Integer'Image (Long_Long_Integer_Get_Hamming (1_000_000)));end Hamming;
Output:
1) First 20 Hamming numbers:  1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36
2) 1_691 st Hamming number:  2125764000

Execution terminated by unhandled exception
Exception name: CONSTRAINT_ERROR
Call stack traceback locations:
0x403212 0x402fd7 0x402a87 0x7f8b99517584 0x4026d7

For using Big_Number, you just have to add this to the code (additional to with Big_Number; and with Ada.Strings.Unbounded; in context clause):

   type My_Index is mod 2**8;   package My_Big_Numbers is new Big_Number (Index_type => My_Index, Nb_Item => 64);   function Int2Big is new My_Big_Numbers.Generic_Conversion.Int_Number2Big_Unsigned (Integer);    function Big_Get_Hamming is new Get_Hamming      (Int_Type => My_Big_Numbers.Big_Unsigned,       Zero     => My_Big_Numbers.Big_Unsigned_Zero,       One      => My_Big_Numbers.Big_Unsigned_One,       Two      => My_Big_Numbers.Big_Unsigned_Two,       Three    => Int2Big(3),       Five     => Int2Big(5),       "mod"    => My_Big_Numbers.Unsigned_Number."mod",       "+"      => My_Big_Numbers.Unsigned_Number."+",       "/"      => My_Big_Numbers.Unsigned_Number."/");

and then use it like this:

   Ada.Text_IO.Put_Line ("3) 1_000_000st Hamming number: " &      Ada.Strings.Unbounded.To_String (My_Big_Numbers.String_Conversion.Big_Unsigned2UString (Big_Get_Hamming (1_000_000))));

## AutoHotkey

SetBatchLines, -1Msgbox % hamming(1,20)Msgbox % hamming(1690)return hamming(first,last=0){	if (first < 1)		ans=ERROR 	if (last = 0)		last := first 	i:=0, j:=0, k:=0 	num1 := ceil((last * 20)**(1/3))	num2 := ceil(num1 * ln(2)/ln(3))	num3 := ceil(num1 * ln(2)/ln(5)) 	loop	{		H := (2**i) * (3**j) * (5**k)		if (H > 0)			ans = %H%n%ans%		i++		if (i > num1)		{			i=0			j++			if (j > num2)			{				j=0				k++			}		}		if (k > num3)			break	}	Sort ans, N 	Loop, parse, ans, n, r 	{		if (A_index > last)			break		if (A_index < first)			continue		Output = %Output%n%A_LoopField%	} 	return Output}

## ALGOL 68

Hamming numbers are generated in a trivial iterative way as in the Python version below. This program keeps the series needed to generate the numbers as short as possible using flexible rows; on the downside, it spends considerable time on garbage collection.

PR precision=100 PR MODE SERIES = FLEX [1 : 0] UNT, # Initially, no elements #     UNT = LONG LONG INT; # A 100-digit unsigned integer # PROC hamming number = (INT n) UNT: # The n-th Hamming number #     CASE n     IN 1, 2, 3, 4, 5, 6, 8, 9, 10, 12 # First 10 in a table #     OUT # Additional operators #         OP MIN = (INT i, j) INT: (i < j | i | j), MIN = (UNT i, j) UNT: (i < j | i | j);         PRIO MIN = 9;         OP LAST = (SERIES h) UNT: h[UPB h]; # Last element of a series #         OP +:= = (REF SERIES s, UNT elem) VOID:            # Extend a series by one element, only keep the elements you need #            (INT lwb = (i MIN j) MIN k, upb = UPB s;              REF SERIES new s = HEAP FLEX [lwb : upb + 1] UNT;             (new s[lwb : upb] := s[lwb : upb], new s[upb + 1] := elem);             s := new s            );         # Determine the n-th hamming number iteratively #         SERIES h := 1, # Series, initially one element #         UNT m2 := 2, m3 := 3, m5 := 5, # Multipliers #         INT i := 1, j := 1, k := 1; # Counters #         TO n - 1         DO h +:= (m2 MIN m3) MIN m5;            (LAST h = m2 | m2 := 2 * h[i +:= 1]);            (LAST h = m3 | m3 := 3 * h[j +:= 1]);            (LAST h = m5 | m5 := 5 * h[k +:= 1])         OD;         LAST h     ESAC; FOR k TO 20DO print ((whole (hamming number (k), 0), blank)) OD;print ((newline, whole (hamming number (1 691), 0)));print ((newline, whole (hamming number (1 000 000), 0)))
Output:
1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36
2125764000
519312780448388736089589843750000000000000000000000000000000000000000000000000000000


## AWK

 # syntax: GAWK -f HAMMING_NUMBERS.AWKBEGIN {    for (i=1; i<=20; i++) {      printf("%d ",hamming(i))    }    printf("\n1691: %d\n",hamming(1691))    exit(0)}function hamming(limit,  h,i,j,k,n,x2,x3,x5) {    h[0] = 1    x2 = 2    x3 = 3    x5 = 5    for (n=1; n<=limit; n++) {      h[n] = min(x2,min(x3,x5))      if (h[n] == x2) { x2 = 2 * h[++i] }      if (h[n] == x3) { x3 = 3 * h[++j] }      if (h[n] == x5) { x5 = 5 * h[++k] }    }    return(h[limit-1])}function min(x,y) {    return((x < y) ? x : y)}

output:

1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36
1691: 2125764000


## BBC BASIC

      @% = &1010      FOR h% = 1 TO 20        PRINT "H("; h% ") = "; FNhamming(h%)      NEXT      PRINT "H(1691) = "; FNhamming(1691)      END       DEF FNhamming(l%)      LOCAL i%, j%, k%, n%, m, x2, x3, x5, h%()      DIM h%(l%) : h%(0) = 1      x2 = 2 : x3 = 3 : x5 = 5      FOR n% = 1 TO l%-1        m = x2        IF m > x3 m = x3        IF m > x5 m = x5        h%(n%) = m        IF m = x2 i% += 1 : x2 = 2 * h%(i%)        IF m = x3 j% += 1 : x3 = 3 * h%(j%)        IF m = x5 k% += 1 : x5 = 5 * h%(k%)      NEXT      = h%(l%-1)

Output:

H(1) = 1
H(2) = 2
H(3) = 3
H(4) = 4
H(5) = 5
H(6) = 6
H(7) = 8
H(8) = 9
H(9) = 10
H(10) = 12
H(11) = 15
H(12) = 16
H(13) = 18
H(14) = 20
H(15) = 24
H(16) = 25
H(17) = 27
H(18) = 30
H(19) = 32
H(20) = 36
H(1691) = 2125764000


## C

Using a min-heap to keep track of numbers. Does not handle big integers.

#include <stdio.h>#include <stdlib.h> typedef unsigned long long ham; size_t alloc = 0, n = 1;ham *q = 0; void qpush(ham h){	int i, j;	if (alloc <= n) {		alloc = alloc ? alloc * 2 : 16;		q = realloc(q, sizeof(ham) * alloc);	} 	for (i = n++; (j = i/2) && q[j] > h; q[i] = q[j], i = j);	q[i] = h;} ham qpop(){	int i, j;	ham r, t;	/* outer loop for skipping duplicates */	for (r = q[1]; n > 1 && r == q[1]; q[i] = t) {		/* inner loop is the normal down heap routine */		for (i = 1, t = q[--n]; (j = i * 2) < n;) {			if (j + 1 < n && q[j] > q[j+1]) j++;			if (t <= q[j]) break;			q[i] = q[j], i = j;		}	} 	return r;} int main(){	int i;	ham h; 	for (qpush(i = 1); i <= 1691; i++) {		/* takes smallest value, and queue its multiples */		h = qpop();		qpush(h * 2);		qpush(h * 3);		qpush(h * 5); 		if (i <= 20 || i == 1691)			printf("%6d: %llu\n", i, h);	} 	/* free(q); */	return 0;}

### Alternative

Standard algorithm. Numbers are stored as exponents of factors instead of big integers, while GMP is only used for display. It's much more efficient this way.

#include <stdio.h>#include <stdlib.h>#include <string.h>#include <math.h>#include <gmp.h> /* number of factors.  best be mutually prime -- duh. */#define NK 3#define MAX_HAM (1 << 24)#define MAX_POW 1024int n_hams = 0, idx[NK] = {0}, fac[] = { 2, 3, 5, 7, 11}; /*  k-smooth numbers are stored as their exponents of each factor;    v is the log of the number, for convenience. */typedef struct {	int e[NK];	double v;} ham_t, *ham; ham_t *hams, values[NK] = {{{0}, 0}};double inc[NK][MAX_POW]; /* most of the time v can be just incremented, but eventually * floating point precision will bite us, so better recalculate */inlinevoid _setv(ham x) {	int i;	for (x->v = 0, i = 0; i < NK; i++)		x->v += inc[i][x->e[i]];} inlineint _eq(ham a, ham b) {	int i;	for (i = 0; i < NK && a->e[i] == b->e[i]; i++); 	return i == NK;} ham get_ham(int n){	int i, ni;	ham h; 	n--;	while (n_hams < n) {		for (ni = 0, i = 1; i < NK; i++)			if (values[i].v < values[ni].v)				ni = i; 		*(h = hams + ++n_hams) = values[ni]; 		for (ni = 0; ni < NK; ni++) {			if (! _eq(values + ni, h)) continue;			values[ni] = hams[++idx[ni]];			values[ni].e[ni]++;			_setv(values + ni);		}	} 	return hams + n;} void show_ham(ham h){	static mpz_t das_ham, tmp;	int i;  	mpz_init_set_ui(das_ham, 1);	mpz_init_set_ui(tmp, 1);	for (i = 0; i < NK; i++) {		mpz_ui_pow_ui(tmp, fac[i], h->e[i]);		mpz_mul(das_ham, das_ham, tmp);	}	gmp_printf("%Zu\n", das_ham);} int main(){	int i, j;	hams = malloc(sizeof(ham_t) * MAX_HAM); 	for (i = 0; i < NK; i++) {		values[i].e[i] = 1;		inc[i][1] = log(fac[i]);		_setv(values + i); 		for (j = 2; j < MAX_POW; j++)			inc[i][j] = j * inc[i][1];	} 	printf("     1,691: "); show_ham(get_ham(1691));	printf(" 1,000,000: "); show_ham(get_ham(1e6));	printf("10,000,000: "); show_ham(get_ham(1e7));	return 0;}
Output:
     1,691: 2125764000
1,000,000: 519312780448388736089589843750000000000000000000000000000000000000000000000000000000
10,000,000: 16244105063830431823239 ..<a gadzillion digits>.. 000000000000000000000

## C++

### C++11 For Each Generator

 #include <iostream>#include <vector>// Hamming like sequences Generator//// Nigel Galloway. August 13th., 2012//class Ham {private:	std::vector<unsigned int> _H, _hp, _hv, _x;public:	bool operator!=(const Ham& other) const {return true;}	Ham begin() const {return *this;}        Ham end() const {return *this;}	unsigned int operator*() const {return _x.back();}	Ham(const std::vector<unsigned int> &pfs):_H(pfs),_hp(pfs.size(),0),_hv({pfs}),_x({1}){}	const Ham& operator++() {	  for (int i=0; i<_H.size(); i++) for (;_hv[i]<=_x.back();_hv[i]=_x[++_hp[i]]*_H[i]);	  _x.push_back(_hv[0]);	  for (int i=1; i<_H.size(); i++) if (_hv[i]<_x.back()) _x.back()=_hv[i];	  return *this;	}};

### 5-Smooth

 int main() {  int count = 1;  for (unsigned int i : Ham({2,3,5})) {    if (count <= 62) std::cout << i << ' ';    if (count++ == 1691) {      std::cout << "\nThe one thousand six hundred and ninety first Hamming Number is " << i << std::endl;      break;    }  }  return 0;}

Produces:

1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36 40 45 48 50 54 60 64 72 75 80 81 90 96 100 108 120 125 128 135 144 150 160 162 180 192 200 216 225 240 243 250 256 270 288 300 320 324 360 375 384 400 405
The one thousand six hundred and ninety first Hamming Number is 2125764000


### 7-Smooth

 int main() {  int count = 1;  for (unsigned int i : Ham({2,3,5,7})) {    std::cout << i << ' ';    if (count++ == 64) break;  }  std::cout << std::endl;  return 0;}

Produces:

1 2 3 4 5 6 7 8 9 10 12 14 15 16 18 20 21 24 25 27 28 30 32 35 36 40 42 45 48 49 50 54 56 60 63 64 70 72 75 80 81 84 90 96 98 100 105 108 112 120 125 126 128 135 140 144 147 150 160 162 168 175 180 189


## C#

Translation of: D
using System;using System.Numerics;using System.Linq; namespace Hamming {     class MainClass {         public static BigInteger Hamming(int n) {            BigInteger two = 2, three = 3, five = 5;            var h = new BigInteger[n];            h[0] = 1;            BigInteger x2 = 2, x3 = 3, x5 = 5;            int i = 0, j = 0, k = 0;             for (int index = 1; index < n; index++) {                h[index] = BigInteger.Min(x2, BigInteger.Min(x3, x5));                if (h[index] == x2) x2 = two * h[++i];                if (h[index] == x3) x3 = three * h[++j];                if (h[index] == x5) x5 = five * h[++k];            }            return h[n - 1];        }         public static void Main(string[] args) {            Console.WriteLine(string.Join(" ", Enumerable.Range(1, 20).ToList().Select(x => Hamming(x))));            Console.WriteLine(Hamming(1691));            Console.WriteLine(Hamming(1000000));        }    }}
Output:
1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36
2125764000
519312780448388736089589843750000000000000000000000000000000000000000000000000000000

### Generic version for any set of numbers

The algorithm is similar to the one above.

using System;using System.Numerics;using System.Linq; namespace Hamming {     class MainClass {         public static BigInteger[] Hamming(int n, int[] a) {            var primes = a.Select(x => (BigInteger)x).ToArray();            var values = a.Select(x => (BigInteger)x).ToArray();            var indexes = new int[a.Length];            var results = new BigInteger[n];            results[0] = 1;            for (int iter = 1; iter < n; iter++) {                results[iter] = values[0];                for (int p = 1; p < primes.Length; p++)                    if (results[iter] > values[p])                        results[iter] = values[p];                for (int p = 0; p < primes.Length; p++)                    if (results[iter] == values[p])                        values[p] = primes[p] * results[++indexes[p]];            }            return results;        }         public static void Main(string[] args) {            foreach (int[] primes in new int[][] { new int[] {2,3,5}, new int[] {2,3,5,7} }) {                Console.WriteLine("{0}-Smooth:", primes.Last());                Console.WriteLine(string.Join(" ", Hamming(20, primes)));                Console.WriteLine(Hamming(1691, primes).Last());                Console.WriteLine(Hamming(1000000, primes).Last());                Console.WriteLine();            }        }    }}
Output:
5-Smooth:
1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36
2125764000
519312780448388736089589843750000000000000000000000000000000000000000000000000000000

7-Smooth:
1 2 3 4 5 6 7 8 9 10 12 14 15 16 18 20 21 24 25 27
3317760
4157409948433216829957008507500000000

### Fast version

Like some of the other implementations on this page, this version represents each number as a list of exponents which would be applied to each prime number. So the number 60 would be represented as int[3] { 2, 1, 1 } which is interpreted as 2^2 * 3^1 * 5^1.

As often happens, optimizing for speed caused a marked increase in code size and complexity. Clearly the versions I wrote above are easier to read & understand. They were also much quicker to write. But the generic version above runs in 3+ seconds for the 1000000th 5-smooth number whereas this version does it in 0.35 seconds, 8-10 times faster.

I've tried to comment it as best I could, without bloating the code too much.

--Mike Lorenz

using System;using System.Linq;using System.Numerics; namespace HammingFast {     class MainClass {         private static int[] _primes = { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 };         public static BigInteger Big(int[] exponents) {            BigInteger val = 1;            for (int i = 0; i < exponents.Length; i++)                for (int e = 0; e < exponents[i]; e++)                    val = val * _primes[i];            return val;        }         public static int[] Hamming(int n, int nprimes) {            var hammings  = new int[n, nprimes];                    // array of hamming #s we generate            var hammlogs  = new double[n];                          // log values for above            var primelogs = new double[nprimes];                    // pre-calculated prime log values            var indexes   = new int[nprimes];                       // intermediate hamming values as indexes into hammings            var listheads = new int[nprimes, nprimes];              // intermediate hamming list heads            var listlogs  = new double[nprimes];                    // log values of list heads            for (int p = 0; p < nprimes; p++) {                listheads[p, p] = 1;                                // init list heads to prime values                primelogs[p]    = Math.Log(_primes[p]);             // pre-calc prime log values                listlogs[p]     = Math.Log(_primes[p]);             // init list head log values            }            for (int iter = 1; iter < n; iter++) {                int min = 0;                                        // find index of min item in list heads                for (int p = 1; p < nprimes; p++)                    if (listlogs[p] < listlogs[min])                        min = p;                hammlogs[iter] = listlogs[min];                     // that's the next hamming number                for (int i = 0; i < nprimes; i++)                    hammings[iter, i] = listheads[min, i];                for (int p = 0; p < nprimes; p++) {                 // update each list head if it matches new value                    bool equal = true;                              // test each exponent to see if number matches                    for (int i = 0; i < nprimes; i++) {                        if (hammings[iter, i] != listheads[p, i]) {                            equal = false;                            break;                        }                    }                    if (equal) {                                    // if it matches...                        int x = ++indexes[p];                       // set index to next hamming number                        for (int i = 0; i < nprimes; i++)           // copy each hamming exponent                            listheads[p, i] = hammings[x, i];                        listheads[p, p] += 1;                       // increment exponent = mult by prime                        listlogs[p] = hammlogs[x] + primelogs[p];   // add log(prime) to log(value) = mult by prime                    }                }            }             var result = new int[nprimes];            for (int i = 0; i < nprimes; i++)                result[i] = hammings[n - 1, i];            return result;        }         public static void Main(string[] args) {            foreach (int np in new int[] { 3, 4, 5 }) {                Console.WriteLine("{0}-Smooth:", _primes[np - 1]);                Console.WriteLine(string.Join(" ", Enumerable.Range(1, 20).Select(x => Big(Hamming(x, np)))));                Console.WriteLine(Big(Hamming(1691, np)));                Console.WriteLine(Big(Hamming(1000000, np)));                Console.WriteLine();            }        }    }}
Output:
5-Smooth:
1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36
2125764000
519312780448388736089589843750000000000000000000000000000000000000000000000000000000

7-Smooth:
1 2 3 4 5 6 7 8 9 10 12 14 15 16 18 20 21 24 25 27
3317760
4157409948433216829957008507500000000

11-Smooth:
1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 18 20 21 22 24
296352
561912530929780078125000

### Old version

This version is interesting because it uses a lazy functional style (using yields), but it's extremely slow; probably two orders of magnitude slower than the other C# versions.

 This example is incomplete. It only shows the first twenty Hamming numbers. This implementation is incapable of calculating the 1691st Hamming number. Please ensure that it meets all task requirements and remove this message.
namespace RosettaCode.HammingNumbers{    using System;    using System.Collections.Generic;    using System.Linq;     internal static class Program    {        private static IEnumerable<int> HammingNumbers        {            get            {                yield return 1;                foreach (var number in                    Merge(HammingNumbers.Select(number => number * 2),                          HammingNumbers.Select(number => number * 3),                          HammingNumbers.Select(number => number * 5)))                {                    yield return number;                }            }        }         private static IEnumerable<int> Merge(params IEnumerable<int>[] sequences)        {            var enumerators = Array.ConvertAll(sequences,                                               sequence => sequence.GetEnumerator());            var minimum = default(int);            while (true)            {                Array.ForEach(enumerators, enumerator =>                                           {                                               if (enumerator.Current == minimum)                                               {                                                   enumerator.MoveNext();                                               }                                           });                yield return minimum = enumerators.Min(enumerator => enumerator.Current);            }        }         private static void Main()        {            foreach (var number in HammingNumbers.Take(20))            {                Console.WriteLine(number);            }        }    }}

## Clojure

This version implements Dijkstra's merge solution, so is closely related to the Haskell version.

(defn smerge [xs ys]  (lazy-seq    (let [x (first xs),          y (first ys),          [z xs* ys*]          (cond            (< x y) [x (rest xs) ys]            (> x y) [y xs (rest ys)]            :else   [x (rest xs) (rest ys)])]      (cons z (smerge xs* ys*))))) (defn smerge3 [xs ys zs]  (smerge xs (smerge ys zs))) (defn map*n [n ks] (map #(*' n %) ks)) (def hamming  (lazy-seq    (cons 1 (smerge3 (map*n 2 hamming) (map*n 3 hamming) (map*n 5 hamming)))))

Note that this version uses a lot of space and time after calculating a few hundred thousand elements of the sequence. This is no doubt due to its "holding on to the head": it maintains the entire generated sequence in memory.

## CoffeeScript

# Generate hamming numbers in order.  Hamming numbers have the# property that they don't evenly divide any prime numbers outside# a given set, such as [2, 3, 5]. generate_hamming_sequence = (primes, max_n) ->  # We use a lazy algorithm, only ever keeping N candidates  # in play, one for each of our seed primes.  Let's say  # primes is [2,3,5].  Our virtual streams are these:  #  # hammings:    1,2,3,4,5,6,8,10,12,15,16,18,20,...  # hammings*2:  2,4,6,9.10,12,16,20,24,30,32,36,40...  # hammings*3:  3,6,9,12,15,18,24,30,36,45,...  # hammings*5:  5,10,15,20,25,30,40,50,...  #  # After encountering 40 for the last time, our candidates   # will be  #   50 = 2 * 25  #   45 = 3 * 15  #   50 = 5 * 10  # Then, after 45  #   50 = 2 * 25  #   48 = 3 * 16 <= new  #   50 = 5 * 10   hamming_numbers = [1]  candidates = ([p, p, 1] for p in primes)  last_number = 1  while hamming_numbers.length < max_n    # Get the next candidate Hamming Number tuple.    i = min_idx(candidates)    candidate = candidates[i]    [n, p, seq_idx] = candidate     # Add to sequence unless it's a duplicate.    if n > last_number      hamming_numbers.push n      last_number = n     # Replace the candidate with its successor (based on    # p = 2, 3, or 5).    #    # This is the heart of the algorithm.  Let's say, over the     # primes [2,3,5], we encounter the hamming number 32 based on it being     # 2 * 16, where 16 is the 12th number in the sequence.    # We'll be passed in [32, 2, 12] as candidate, and    # hamming_numbers will be [1,2,3,4,5,6,8,9,10,12,16,18,...]    # by now.  The next candidate we need to enqueue is    # [36, 2, 13], where the numbers mean this:    #    #    36 - next multiple of 2 of a Hamming number    #     2 - prime number    #    13 - 1-based index of 18 in the sequence    #     # When we encounter [36, 2, 13], we will then enqueue    # [40, 2, 14], based on 20 being the 14th hamming number.    q = hamming_numbers[seq_idx]    candidates[i] = [p*q, p, seq_idx+1]   hamming_numbers min_idx = (arr) ->  # Don't waste your time reading this--it just returns  # the index of the smallest tuple in an array, respecting that  # the tuples may contain integers. (CS compiles to JS, which is  # kind of stupid about sorting.  There are libraries to work around  # the limitation, but I wanted this code to be standalone.)  less_than = (tup1, tup2) ->    i = 0    while i < tup2.length      return true if tup1[i] <= tup2[i]      return false if tup1[i] > tup2[i]      i += 1   min_i = 0  for i in [1...arr.length]    if less_than arr[i], arr[min_i]      min_i = i  return min_i primes = [2, 3, 5]numbers = generate_hamming_sequence(primes, 10000)console.log numbers[1690]console.log numbers[9999]

## Common Lisp

Maintaining three queues, popping the smallest value every time.

(defun next-hamm (factors seqs)  (let ((x (apply #'min (map 'list #'first seqs))))    (loop for s in seqs	  for f in factors	  for i from 0	  with add = t do	  (if (= x (first s)) (pop s))	  ;; prevent a value from being added to multiple lists	  (when add	    (setf (elt seqs i) (nconc s (list (* x f))))	    (if (zerop (mod x f)) (setf add nil)))    finally (return x)))) (loop with factors = '(2 3 5)      with seqs    = (loop for i in factors collect '(1))      for n from 1 to 1000001 do      (let ((x (next-hamm factors seqs)))	(if (or (< n 21)		(= n 1691)		(= n 1000000)) (format t "~d: ~d~%" n x))))

A much faster method:

(defun hamming (n)  (let ((fac '(2 3 5))	(idx (make-array 3 :initial-element 0))	(h (make-array (1+ n)		       :initial-element 1		       :element-type 'integer)))    (loop for i from 1 to n	  with e with x = '(1 1 1) do	  (setf e (setf (aref h i) (apply #'min x))		x (loop for y in x			for f in fac			for j from 0			collect (if (= e y) (* f (aref h (incf (aref idx j)))) y))))    (aref h n))) (loop for i from 1 to 20 do      (format t "~2d: ~d~%" i (hamming i))) (loop for i in '(1691 1000000) do      (format t "~d: ~d~%" i (hamming i)))
Output:
 1: 1
2: 2
3: 3
4: 4
5: 5
6: 6
7: 8
8: 9
9: 10
10: 12
11: 15
12: 16
13: 18
14: 20
15: 24
16: 25
17: 27
18: 30
19: 32
20: 36
1691: 2125764000
1000000: 519312780448388736089589843750000000000000000000000000000000000000000000000000000000

## D

This version keeps all numbers in memory, computing all the Hamming numbers up to the needed one. Performs constant number of operations per Hamming number produced.

import std.stdio, std.bigint, std.algorithm, std.range, core.memory; auto hamming(in int n) {    BigInt two = 2, three = 3, five = 5;    auto h = new BigInt[n];    h[0] = 1;    BigInt x2 = 2, x3 = 3, x5 = 5;    int i, j, k;     foreach (ref el; h[1 .. $]) { el = min(x2, x3, x5); if (el == x2) x2 = two * h[++i]; if (el == x3) x3 = three * h[++j]; if (el == x5) x5 = five * h[++k]; } return h.back;} void main() { GC.disable; iota(1, 21).map!hamming.writeln; 1_691.hamming.writeln; 1_000_000.hamming.writeln;} Output: [1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36] 2125764000 519312780448388736089589843750000000000000000000000000000000000000000000000000000000 Runtime is about 1.6 seconds with LDC2. ### Alternative version 1 This keeps O(n2 / 3) numbers in memory, but over-computes a sequence by a factor of about Θ(n2 / 3), calculating extra multiples past that as well. Incurs an extra O(logn) factor of operations per each number produced (reinserting its multiples into a tree). Doesn't stop when the target number is reached, instead continuing until it is no longer needed: Translation of: Java import std.stdio, std.bigint, std.container, std.algorithm, std.range, core.memory; BigInt hamming(in int n)in { assert(n > 0);} body { auto frontier = redBlackTree(2.BigInt, 3.BigInt, 5.BigInt); auto lowest = 1.BigInt; foreach (immutable _; 1 .. n) { lowest = frontier.front; frontier.removeFront; frontier.insert(lowest * 2); frontier.insert(lowest * 3); frontier.insert(lowest * 5); } return lowest;} void main() { GC.disable; writeln("First 20 Hamming numbers: ", iota(1, 21).map!hamming); writeln("hamming(1691) = ", 1691.hamming); writeln("hamming(1_000_000) = ", 1_000_000.hamming);} Output: First 20 Hamming numbers: [1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36] hamming(1691) = 2125764000 hamming(1_000_000) = 519312780448388736089589843750000000000000000000000000000000000000000000000000000000 About 3.2 seconds run time with LDC2. ### Alternative version 2 Does exactly what the first version does, creating an array and filling it with Hamming numbers, keeping the three back pointers into the sequence for next multiples calculations, except that it represents the numbers as their coefficients triples and their logarithm values (for comparisons), thus saving on BigInt calculations. Translation of: C import std.stdio: writefln;import std.bigint: BigInt;import std.conv: text;import std.numeric: gcd;import std.algorithm: copy, map;import core.stdc.stdlib: calloc;import std.math: log; // ^^ // Number of factors.enum NK = 3; enum MAX_HAM = 10_000_000;static assert(gcd(NK, MAX_HAM) == 1); enum int[NK] factors = [2, 3, 5]; /// K-smooth numbers (stored as their exponents of each factor).struct Hamming { double v; // Log of the number, for convenience. ushort[NK] e; // Exponents of each factor. // log can't be used in CTFE yet. //public static __gshared immutable double[factors.length] inc = // factors[].map!log.array; public static __gshared immutable double[factors.length] inc; nothrow pure static this() { //factors[].map!log.copy(inc[]); // Not nothrow, not const. foreach (immutable i, immutable f; factors) inc[i] = f.log; } bool opEquals(in ref Hamming y) const pure nothrow { //return this.e == y.e; // Too much slow. foreach (immutable i; 0 .. this.e.length) if (this.e[i] != y.e[i]) return false; return true; } void update() pure nothrow { //this.v = dotProduct(inc, this.e); // Too much slow. this.v = 0.0; foreach (immutable i; 0 .. this.e.length) this.v += inc[i] * this.e[i]; } string toString() const { BigInt result = 1; foreach (immutable i, immutable f; factors) result *= f.BigInt ^^ this.e[i]; return result.text; }} // Global variables.__gshared Hamming[] hams;__gshared Hamming[NK] values; nothrow static this() { // Slower than calloc if you don't use all the MAX_HAM items. //hams = new Hamming[MAX_HAM]; auto ptr = cast(Hamming*)calloc(MAX_HAM, Hamming.sizeof); if (!ptr) throw new Error("Not enough memory."); hams = ptr[0 .. MAX_HAM]; foreach (immutable i, ref v; values) { v.e[i] = 1; v.v = Hamming.inc[i]; }} ref Hamming getHam(in size_t n) nothrowin { assert(n <= MAX_HAM);} body { // Most of the time v can be just incremented, but eventually // floating point precision will bite us, so better recalculate. __gshared static size_t[NK] idx; __gshared static int n_hams; for (; n_hams < n; n_hams++) { { // Find the index of the minimum v. size_t ni = 0; foreach (immutable i; 1 .. NK) if (values[i].v < values[ni].v) ni = i; hams[n_hams] = values[ni]; hams[n_hams].update; } foreach (immutable i; 0 .. NK) if (values[i] == hams[n_hams]) { values[i] = hams[idx[i]]; idx[i]++; values[i].e[i]++; values[i].update; } } return hams[n - 2];} void main() { foreach (immutable n; [1691, 10 ^^ 6, MAX_HAM]) writefln("%8d: %s", n, n.getHam);} The output is similar to the second C version. Runtime is about 0.11 seconds if MAX_HAM = 1_000_000 (as the task requires), and 0.90 seconds if MAX_HAM = 10_000_000. ## Eiffel  note description : "Initial part, in order, of the sequence of Hamming numbers" math : "[ Hamming numbers, also known as regular numbers and 5-smooth numbers, are natural integers that have 2, 3 and 5 as their only prime factors. ]" computer_arithmetic : "[ This version avoids integer overflow and stops at the last representable number in the sequence. ]" output : "[ Per requirements of the RosettaCode example, execution will produce items of indexes 1 to 20 and 1691. The algorithm (procedure hamming') is more general and will produce the first n' Hamming numbers for any n'. ]" source : "This problem was posed in Edsger W. Dijkstra, A Discipline of Programming, Prentice Hall, 1978" date : "8 August 2012" authors : "Bertrand Meyer", "Emmanuel Stapf" revision : "1.0" libraries : "Relies on SORTED_TWO_WAY_LIST from EiffelBase" implementation : "[ Using SORTED_TWO_WAY_LIST provides an elegant illustration of how to implement a lazy scheme in Eiffel through the use of object-oriented data structures. ]" warning : "[ The formatting (<lang>) specifications for Eiffel in RosettaCode are slightly obsolete: note' and other newer keywords not supported, red color for manifest strings. This should be fixed soon. ]" class APPLICATION create make feature {NONE} -- Initialization make -- Print first 20 Hamming numbers, in order, and the 1691-st one. local Hammings: like hamming -- List of Hamming numbers, up to 1691-st one. do Hammings := hamming (1691) across 1 |..| 20 as i loop io.put_natural (Hammings.i_th (i.item)); io.put_string (" ") end io.put_new_line; io.put_natural (Hammings.i_th (1691)); io.put_new_line end feature -- Basic operations hamming (n: INTEGER): ARRAYED_LIST [NATURAL] -- First n' elements (in order) of the Hamming sequence, -- or as many of them as will not produce overflow. local sl: SORTED_TWO_WAY_LIST [NATURAL] overflow: BOOLEAN first, next: NATURAL do create Result.make (n); create sl.make sl.extend (1); sl.start across 1 |..| n as i invariant -- "The numbers output so far are the first i' - 1 Hamming numbers, in order". -- "Result.first is the i'-th Hamming number." until sl.is_empty loop first := sl.first; sl.start Result.extend (first); sl.remove across << 2, 3, 5 >> as multiplier loop next := multiplier.item * first overflow := overflow or next <= first if not overflow and then not sl.has (next) then sl.extend (next) end end end endend  Output: 1 1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36 2125764000 ## Factor Translation of: Scala USING: accessors deques dlists fry kernel make math math.order;IN: rosetta.hamming TUPLE: hamming-iterator 2s 3s 5s ; : <hamming-iterator> ( -- hamming-iterator ) hamming-iterator new 1 1dlist >>2s 1 1dlist >>3s 1 1dlist >>5s ; : enqueue ( n hamming-iterator -- ) [ [ 2 * ] [ 2s>> ] bi* push-back ] [ [ 3 * ] [ 3s>> ] bi* push-back ] [ [ 5 * ] [ 5s>> ] bi* push-back ] 2tri ; : next ( hamming-iterator -- n ) dup [ 2s>> ] [ 3s>> ] [ 5s>> ] tri 3dup [ peek-front ] tri@ min min [ '[ dup peek-front _ = [ pop-front* ] [ drop ] if ] tri@ ] [ swap enqueue ] [ ] tri ; : next-n ( hamming-iterator n -- seq ) swap '[ _ [ _ next , ] times ] { } make ; : nth-from-now ( hamming-iterator n -- m ) 1 - over '[ _ next drop ] times next ;  <hamming-iterator> 20 next-n . <hamming-iterator> 1691 nth-from-now . <hamming-iterator> 1000000 nth-from-now .  Translation of: Haskell Lazy lists aren quite slow in Factor, but still. USING: combinators fry kernel lists lists.lazy locals math ;IN: rosetta.hamming-lazy :: sort-merge ( xs ys -- result ) xs car :> x ys car :> y { { [ x y < ] [ [ x ] [ xs cdr ys sort-merge ] lazy-cons ] } { [ x y > ] [ [ y ] [ ys cdr xs sort-merge ] lazy-cons ] } [ [ x ] [ xs cdr ys cdr sort-merge ] lazy-cons ] } cond ; :: hamming ( -- hamming ) f :> h! [ 1 ] [ h 2 3 5 [ '[ _ * ] lazy-map ] tri-curry@ tri sort-merge sort-merge ] lazy-cons h! h ;  20 hamming ltake list>array . 1690 hamming lnth . 999999 hamming lnth .  ## Forth Works with: Gforth version 0.7.0 This version uses a compact representation of Hamming numbers: each 64-bit cell represents a number 2^l*3^m*5^n, where l, n, and m are bitfields in the cell (20 bits for now). It also uses a fixed-point logarithm to compare the Hamming numbers and prints them in factored form. This code has been tested up to the 10^9th Hamming number. \ manipulating and computing with Hamming numbers: : extract2 ( h -- l ) 40 rshift ; : extract3 ( h -- m ) 20 rshift$fffff and ; : extract5 ( h -- n )    $fffff and ; ' + alias h* ( h1 h2 -- h ) : h. { h -- } ." 2^" h extract2 0 .r ." *3^" h extract3 0 .r ." *5^" h extract5 . ; \ the following numbers have been produced with bc -l as follows1 62 lshift constant ldscale2 7309349404307464679 constant ldscale3 \ 2^62*l(3)/l(2) (rounded up)10708003330985790206 constant ldscale5 \ 2^62*l(5)/l(2) (rounded down) : hld { h -- ud } \ ud is a scaled fixed-point representation of the logarithm dualis of h h extract2 ldscale2 um* h extract3 ldscale3 um* d+ h extract5 ldscale5 um* d+ ; : h<= ( h1 h2 -- f ) 2dup = if 2drop true exit then hld rot hld assert( 2over 2over d<> ) du>= ; : hmin ( h1 h2 -- h ) 2dup h<= if drop else nip then ; \ actual algorithm 0 value seqvariable seqlast 0 seqlast ! : lastseq ( -- u ) \ last stored number in the sequence seq seqlast @ th @ ; : genseq ( h1 "name" -- ) \ h1 is the factor for the sequence create , 0 , \ factor and index of element used for last return does> ( -- u2 ) \ u2 is the next number resulting from multiplying h1 with numbers \ in the sequence that is larger than the last number in the \ sequence dup @ lastseq { h1 l } cell+ dup @ begin ( index-addr index ) seq over th @ h1 h* dup l h<= while drop 1+ repeat >r swap ! r> ;$10000000000 genseq s2$00000100000 genseq s3$00000000001 genseq s5 : nextseq ( -- )    s2 s3 hmin s5 hmin , 1 seqlast +! ; : nthseq ( u1 -- h )    \ the u1 th element in the sequence    dup seqlast @ u+do        nextseq    loop    1- 0 max cells seq + @ ; : .nseq ( u1 -- )    dup seqlast @ u+do        nextseq    loop    0 u+do        seq i th @ h.    loop ; here to seq0 , \ that's 1 20 .nseqcr    1691 nthseq h.cr 1000000 nthseq h.
Output:
2^0*3^0*5^0 2^1*3^0*5^0 2^0*3^1*5^0 2^2*3^0*5^0 2^0*3^0*5^1 2^1*3^1*5^0 2^3*3^0*5^0 2^0*3^2*5^0 2^1*3^0*5^1 2^2*3^1*5^0 2^0*3^1*5^1 2^4*3^0*5^0 2^1*3^2*5^0 2^2*3^0*5^1 2^3*3^1*5^0 2^0*3^0*5^2 2^0*3^3*5^0 2^1*3^1*5^1 2^5*3^0*5^0 2^2*3^2*5^0
2^5*3^12*5^3
2^55*3^47*5^64


## Fortran

Works with: Fortran version 90 and later

Using big_integer_module from here[1]

program Hamming_Test  use big_integer_module  implicit none   call Hamming(1,20)  write(*,*)  call Hamming(1691)  write(*,*)  call Hamming(1000000) contains subroutine Hamming(first, last)   integer, intent(in) :: first  integer, intent(in), optional :: last  integer :: i, n, i2, i3, i5, lim  type(big_integer), allocatable :: hnums(:)   if(present(last)) then    lim = last  else    lim = first  end if   if(first < 1 .or. lim > 2500000 ) then    write(*,*) "Invalid input"    return  end if   allocate(hnums(lim))   i2 = 1 ;  i3 = 1 ; i5 = 1    hnums(1) = 1  n = 1  do while(n < lim)    n = n + 1    hnums(n) = mini(2*hnums(i2), 3*hnums(i3), 5*hnums(i5))    if(2*hnums(i2) == hnums(n)) i2 = i2 + 1    if(3*hnums(i3) == hnums(n)) i3 = i3 + 1    if(5*hnums(i5) == hnums(n)) i5 = i5 + 1  end do   if(present(last)) then    do i = first, last      call print_big(hnums(i))      write(*, "(a)", advance="no") " "    end do  else    call print_big(hnums(first))  end if   deallocate(hnums)end subroutine function mini(a, b, c)  type(big_integer) :: mini  type(big_integer), intent(in) :: a, b, c   if(a < b ) then     if(a < c) then      mini = a    else      mini = c    end if  else if(b < c) then    mini = b  else    mini = c  end if end function mini end program
Output:
1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36
2125764000
519312780448388736089589843750000000000000000000000000000000000000000000000000000000

## Go

### Concise version using dynamic-programming

package main import (    "fmt"    "math/big") func min(a, b *big.Int) *big.Int {    if a.Cmp(b) < 0 {        return a    }    return b} func hamming(n int) []*big.Int {    h := make([]*big.Int, n)    h[0] = big.NewInt(1)    two, three, five    := big.NewInt(2), big.NewInt(3), big.NewInt(5)    next2, next3, next5 := big.NewInt(2), big.NewInt(3), big.NewInt(5)    i, j, k := 0, 0, 0    for m := 1; m < len(h); m++ {        h[m] = new(big.Int).Set(min(next2, min(next3, next5)))        if h[m].Cmp(next2) == 0 { i++; next2.Mul(  two, h[i]) }         if h[m].Cmp(next3) == 0 { j++; next3.Mul(three, h[j]) }         if h[m].Cmp(next5) == 0 { k++; next5.Mul( five, h[k]) }     }    return h} func main() {    h := hamming(1e6)    fmt.Println(h[:20])    fmt.Println(h[1691-1])    fmt.Println(h[len(h)-1])}
Output:
[1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36]
2125764000
519312780448388736089589843750000000000000000000000000000000000000000000000000000000


### Longer version using dynamic-programming and logarithms

More than 10 times faster.

package main import (    "flag"    "fmt"    "math"    "math/big"    "os") var ordinal int       // ordinal of last sequence element to computevar sequenceMode bool // print the whole sequence or just one element? var lg3, lg5 float64 // precomputed base-2 logarithms for 3 and 5 var table [][3]int16 // table for dynamic-programming stored resultsvar front [3]cursor  // state of the three multiplied sequences type cursor struct {    f  int     // index (0, 1, 2) corresponding to factor (2, 3, 5)    i  int     // index into table for the entry being multiplied    lg float64 // base-2 logarithm of the multiple (for ordering)} func (c *cursor) value() [3]int16 {    x := table[c.i]    x[c.f]++ // multiply by incrementing the exponent    return x} func (c *cursor) advance() {    c.i++    // skip entries that would produce duplicates    for (c.f < 2 && table[c.i][2] > 0) || (c.f < 1 && table[c.i][1] > 0) {        c.i++    }    x := c.value()    c.lg = float64(x[0]) + lg3*float64(x[1]) + lg5*float64(x[2])} func step() {    table = append(table, front[0].value())    front[0].advance()    // re-establish sorted order    if front[0].lg > front[1].lg {        front[0], front[1] = front[1], front[0]        if front[1].lg > front[2].lg {            front[1], front[2] = front[2], front[1]        }    }}func show(elem [3]int16) {    z := big.NewInt(1)    for i, base := range []int64{2, 3, 5} {        b := big.NewInt(base)        x := big.NewInt(int64(elem[i]))        z.Mul(z, b.Exp(b, x, nil))    }    fmt.Println(z)} func fail(msg string) {    fmt.Fprintf(os.Stderr, "%s: %s\n", os.Args[0], msg)    os.Exit(1)} func parse() {    flag.Parse()    if flag.NArg() != 1 {        fail("need one argument")    }    _, err := fmt.Sscan(flag.Arg(0), &ordinal)    if err != nil || ordinal <= 0 {        fail("argument must be a positive integer")    }} func init() {    flag.BoolVar(&sequenceMode, "s", false, "sequence mode")    lg3 = math.Log2(3)    lg5 = math.Log2(5)    front = [3]cursor{        {0, 0, 1},   // 2        {1, 0, lg3}, // 3        {2, 0, lg5}, // 5    }} func main() {    parse()    table = make([][3]int16, 1, ordinal)    for i, n := 1, ordinal; i < n; i++ {        if sequenceMode {            show(table[i-1])        }        step()    }    show(table[ordinal-1])}
Output:
$./hamming -s 20 | xargs 1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36$ time ./hamming 1000000
519312780448388736089589843750000000000000000000000000000000000000000000000000000000

real	0m0.110s
user	0m0.090s
sys	0m0.020s

### Explicit multiples reinserting

This is a common approach which explicitly maintains an internal buffer of O(n2 / 3) elements, removing the numbers from its front and reinserting their 2- 3- and 5-multiples in order. It overproduces the sequence, stopping when the n-th number is no longer needed instead of when it's first found. Also overworks by maintaining this buffer in total order where just heap would be sufficient. Worse, this particular version uses a sequential list for its buffer. That means O(n2 / 3) operations for each number, instead of O(1) of the above version. Translation of Java (which does use priority queue though, so should have O‍ ‍(n‍ ‍logn) operations overall). Uses union from the "classic" version above:

hamm n = drop n $iterate (\(_,(a:t))-> (a,union t [2*a,3*a,5*a])) (0,[1]) Output: *Main> map fst$ take 20 $hamm 1[1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,27,30,32,36] *Main> map fst$ take 2 $hamm 1691[2125764000,2147483648] *Main> mapM_ print$ take 10 $hamm 1(1,[2,3,5])(2,[3,4,5,6,10])(3,[4,5,6,9,10,15])(4,[5,6,8,9,10,12,15,20])(5,[6,8,9,10,12,15,20,25])(6,[8,9,10,12,15,18,20,25,30])(8,[9,10,12,15,16,18,20,24,25,30,40])(9,[10,12,15,16,18,20,24,25,27,30,40,45])(10,[12,15,16,18,20,24,25,27,30,40,45,50])(12,[15,16,18,20,24,25,27,30,36,40,45,50,60]) *Main> map (length.snd.head.hamm) [2000,4000,8000,16000][402,638,1007,1596] Runs too slowly to reach 1,000,000, with orders of growth above ~‍ ‍(n‍ ‍1.7‍ ‍) and worsening. Last two lines show the internal buffer's length for several sample n‍ ‍s. ### Direct calculation through triples enumeration It is also possible to more or less directly calculate the n-th Hamming number by enumerating (and counting) all the (i,j,k) triples below its estimated value – with ordering according to their exponents, i*ln2 + j*ln3 + k*ln5 – while storing only the "band" of topmost triples close enough to the target value (more in the original post on DDJ). The total count of the enumerated triples is then the band's topmost value's position in the Hamming sequence, 1-based. The nth number is then found by a simple lookup in the sorted band, if it was wide enough. This produces the 1,000,000-th value in a few hundredths of a second on Ideone.com, running at about O(n0.7) empirical time and space complexity: -- directly find n-th Hamming number, in ~ O(n^{2/3}) time-- by Will Ness, based on "top band" idea by Louis Klauder, from DDJ discussion-- http://drdobbs.com/blogs/architecture-and-design/228700538 {-# OPTIONS -O2 -XBangPatterns #-}import Data.List (sortBy)import Data.Function (on) main = let (r,t) = nthHam 1000000 in print t >> print (trival t) lg3 = logBase 2 3; lg5 = logBase 2 5logval (i,j,k) = fromIntegral i + fromIntegral j*lg3 + fromIntegral k*lg5trival (i,j,k) = 2^i * 3^j * 5^kestval n = (6*lg3*lg5* fromIntegral n)**(1/3) -- estimated logval, base 2rngval n | n > 500000 = (2.4496 , 0.0076 ) -- empirical estimation | n > 50000 = (2.4424 , 0.0146 ) -- correction, base 2 | n > 500 = (2.3948 , 0.0723 ) -- (dist,width) | n > 1 = (2.2506 , 0.2887 ) -- around (log$ sqrt 30),     | otherwise   = (2.2506 , 0.5771 )                  --   says WP nthHam :: Int -> (Double, (Int, Int, Int))nthHam n                                                   -- n: 1-based: 1,2,3...  | w >= 1   = error $"Breach of contract: (w < 1): " ++ show w | m < 0 = error$ "Not enough triples generated: " ++ show (c,n)  | m >= nb  = error $"Generated band is too narrow: " ++ show (m,nb) | True = res where (d,w) = rngval n -- correction dist, width hi = estval n - d -- hi > logval > hi-w (m,nb) = ( fromIntegral$ c - n, length b )           -- m 0-based from top, |band|  (s,res) = ( sortBy (flip compare on fst) b, s!!m )   -- sorted decreasing, result  (c,b)   = f 0                                          -- total count, the band              [ ( i+1,                                   -- total triples w/ this (j,k)                  [ (r,(i,j,k)) | frac < w ] )           -- store it, if inside band                | k <- [ 0 .. floor ( hi   /lg5) ],  let p = fromIntegral k*lg5,                  j <- [ 0 .. floor ((hi-p)/lg3) ],  let q = fromIntegral j*lg3 + p,                  let (i,frac) = pr  (hi-q) ;            r = hi-frac ] -- r = i + q         -- f 0 z == (sum $map fst z, concat$ map snd z)    where pr = properFraction          f !c []          = (c,[])                      -- code as a loop           f !c ((c1,b1):r) = let (cr,br) = f (c+c1) r    --   to prevent space leak                             in case b1 of { [v] -> (cr,v:br)                                            ;  _  -> (cr, br) }
Output:
-- time: 0.01s    memory: 3688 kB
(55,47,64)
519312780448388736089589843750000000000000000000000000000000000000000000000000000000

## Icon and Unicon

This solution uses Unicon's object oriented extensions. An Icon only version has not been provided.

Lazy evaluation is used to improve performance.

# Lazily generate the three Hamming numbers that can be derived directly#   from a known Hamming number hclass Triplet : Class (cv, ce)     method nextVal()        suspend cv := @ce    end     initially (baseNum)        cv := 2*baseNum        ce := create (3|5)*baseNumend # Generate Hamming numbers, in order.  Default is first 30#  But an optional argument can be used to generate more (or less)#   e.g. hamming 5000 generates the first 5000.procedure main(args)    limit := integer(args[1]) | 30    every write("\t", generateHamming() \ limit)end # Do the work.   Start with known Hamming number 1 and maintain#   a set of triplet Hamming numbers as they get derived from that#   one.  Most of the code here is to figure out which Hamming#   number is next in sequence (while removing duplicates)procedure generateHamming()    triplers := set()    insert(triplers, Triplet(1))     suspend 1    repeat {        # Pick a Hamming triplet that *may* have the next smallest number        t1 := !triplers         # any will do to start         every t1 ~=== (t2 := !triplers) do {            if t1.cv > t2.cv then {               # oops we were wrong, switch assumption               t1 := t2               }            else if t1.cv = t2.cv then {               # t2's value is a duplicate, so               # advance triplet t2, if none left in t2, remove it               t2.nextVal() | delete(triplers, t2)               }           }         # Ok, t1 has the next Hamming number, grab it        suspend t1.cv        insert(triplers, Triplet(t1.cv))         # Advance triplet t1, if none left in t1, remove it        t1.nextVal() | delete(triplers, t1)        }end

## J

Solution:
A concise tacit expression using a (right) fold:

hamming=: {. (/:~@~.@], 2 3 5 * {)/ @ (1x,~i.@-)

Example usage:

   hamming 201 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36    {: hamming 16912125764000

For the millionth (and billionth (1e9)) Hamming number see the hn verb from Hamming Number essay on the J wiki.

Explanation:
I'll explain this J-sentence by dividing it in three parts from left to right omitting the leftmost {.:

• sort and remove duplicates
 /:~@~.@]
• produce (the next) 3 elements by selection and multiplication:
 2 3 5 * {

or

 2 3 5 * LHA { RHA
• the RH part forms an array of descending indices and the initial Hamming number 1
 (1x,~i.@-)

e.g. if we want the first 5 Hamming numbers, it produces the array:

4 3 2 1 0 1


in other words, we compute a sequence which begins with the desired hamming sequence and then take the first n elements (which will be our desired hamming sequence)

   ({. (/:~@~.@], 2 3 5 * {)/ @ (1x,~i.@-)) 71 2 3 4 5 6 8

This starts using a descending sequence with 1 appended:

    (1x,~i.@-) 76 5 4 3 2 1 0 1

and then the fold expression is inserted between these list elements and the result computed:

   6(/:~@~.@], 2 3 5 * {) 5(/:~@~.@], 2 3 5 * {) 4(/:~@~.@], 2 3 5 * {) 3(/:~@~.@], 2 3 5 * {) 2(/:~@~.@], 2 3 5 * {) 1(/:~@~.@], 2 3 5 * {) 0(/:~@~.@], 2 3 5 * {) 11 2 3 4 5 6 8 9 10 12 15 18 20 25 30 16 24 40

(Note: A train of verbs in J is evaluated by supplying arguments to the every other verb (counting from the right) and the combining these results with the remaining verbs. Also: { has been implemented so that an index of 0 will select the only item from an array with no dimensions.)

## Java

Works with: Java version 1.5+

Has a common shortcoming of overproducing the sequence by about O(n2 / 3) entries, until the n-th number is no longer needed, instead of stopping as soon as it is reached. See Haskell for an illustration.

Inserting the top number's three multiples deep into the priority queue as it does, incurs extra cost for each number produced. To not worsen the expected algorithm complexity, the priority queue should have (amortized) O(1) implementation for both insertion and deletion operations but it looks like it's O(logn) in Java.

import java.math.BigInteger;import java.util.PriorityQueue; final class Hamming {    private static BigInteger THREE = BigInteger.valueOf(3);    private static BigInteger FIVE = BigInteger.valueOf(5);     private static void updateFrontier(BigInteger x,                                       PriorityQueue<BigInteger> pq) {        pq.offer(x.shiftLeft(1));        pq.offer(x.multiply(THREE));        pq.offer(x.multiply(FIVE));    }     public static BigInteger hamming(int n) {        if (n <= 0)            throw new IllegalArgumentException("Invalid parameter");        PriorityQueue<BigInteger> frontier = new PriorityQueue<BigInteger>();        updateFrontier(BigInteger.ONE, frontier);        BigInteger lowest = BigInteger.ONE;        for (int i = 1; i < n; i++) {            lowest = frontier.poll();            while (frontier.peek().equals(lowest))                frontier.poll();            updateFrontier(lowest, frontier);        }        return lowest;    }     public static void main(String[] args) {        System.out.print("Hamming(1 .. 20) =");        for (int i = 1; i < 21; i++)             System.out.print(" " + hamming(i));        System.out.println("\nHamming(1691) = " + hamming(1691));        System.out.println("Hamming(1000000) = " + hamming(1000000));    }}
Output:
Hamming(1 .. 20) = 1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36
Hamming(1691) = 2125764000
Hamming(1000000) = 519312780448388736089589843750000000000000000000000000000000000000000000000000000000

## JavaScript

Works with: JavaScript version 1.7
Works with: Firefox version 2
Translation of: Ruby

This does not calculate the 1,000,000th Hamming number.

Note the use of for (x in obj) to iterate over the properties of an object, versus for each (y in obj) to iterate over the values of the properties of an object.

function hamming() {    var queues = {2: [], 3: [], 5: []};    var base;    var next_ham = 1;    while (true) {        yield next_ham;         for (base in queues) {queues[base].push(next_ham * base)}         next_ham = [ queue[0] for each (queue in queues) ].reduce(function(min, val) {            return Math.min(min,val)        });         for (base in queues) {if (queues[base][0] == next_ham) queues[base].shift()}    }} var ham = hamming();var first20=[], i=1; for (; i <= 20; i++)     first20.push(ham.next());print(first20.join(', '));print('...');for (; i <= 1690; i++)     ham.next();print(i + " => " + ham.next());
Output:
1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36
...
1691 => 2125764000 

### Fast & complete version

Translation of: C#

A translation of my fast C# version. I was curious to see how much slower JavaScript is. The result: it runs about 5x times slower than C#, though YMMV. You can try it yourself here: http://jsfiddle.net/N7AFN/

--Mike Lorenz

<html><head></head><body>    <div id="main"></div></body><script src="http://code.jquery.com/jquery-latest.min.js"></script><script src="http://peterolson.github.com/BigInteger.js/BigInteger.min.js"></script><script type="text/javascript">    var _primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37];     function log(text) {        $('#main').append(text + "\n"); } function big(exponents) { var i, e, val = bigInt.one; for (i = 0; i < exponents.length; i++) for (e = 0; e < exponents[i]; e++) val = val.times(_primes[i]); return val.toString(); } function hamming(n, nprimes) { var i, iter, p, q, min, equal, x; var hammings = new Array(n); // array of hamming #s we generate hammings[0] = new Array(nprimes); for (p = 0; p < nprimes; p++) { hammings[0][p] = 0; } var hammlogs = new Array(n); // log values for above hammlogs[0] = 0; var primelogs = new Array(nprimes); // pre-calculated prime log values var listlogs = new Array(nprimes); // log values of list heads for (p = 0; p < nprimes; p++) { primelogs[p] = listlogs[p] = Math.log(_primes[p]); } var indexes = new Array(nprimes); // intermediate hamming values as indexes into hammings for (p = 0; p < nprimes; p++) { indexes[p] = 0; } var listheads = new Array(nprimes); // intermediate hamming list heads for (p = 0; p < nprimes; p++) { listheads[p] = new Array(nprimes); for (q = 0; q < nprimes; q++) { listheads[p][q] = 0; } listheads[p][p] = 1; } for (iter = 1; iter < n; iter++) { min = 0; for (p = 1; p < nprimes; p++) if (listlogs[p] < listlogs[min]) min = p; hammlogs[iter] = listlogs[min]; // that's the next hamming number hammings[iter] = listheads[min].slice(); for (p = 0; p < nprimes; p++) { // update each list head if it matches new value equal = true; // test each exponent to see if number matches for (i = 0; i < nprimes; i++) { if (hammings[iter][i] != listheads[p][i]) { equal = false; break; } } if (equal) { // if it matches... x = ++indexes[p]; // set index to next hamming number listheads[p] = hammings[x].slice(); // copy hamming number listheads[p][p] += 1; // increment exponent = mult by prime listlogs[p] = hammlogs[x] + primelogs[p]; // add log(prime) to log(value) = mult by prime } } } return hammings[n - 1]; }$(document).ready(function() {        var i, nprimes;        var t = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,1691,1000000];         for (nprimes = 3; nprimes <= 4; nprimes++) {            var start = new Date();            log('<h1>' + _primes[nprimes - 1] + '-Smooth:' + '</h1>');            log('<table>');            for (i = 0; i < t.length; i++)                log('<tr>' + '<td>' + t[i] + ':' + '</td><td>' + big(hamming(t[i], nprimes)) + '</td>');            var end = new Date();            log('<tr>' + '<td>' + 'Elapsed time:' + '</td><td>' + (end-start)/1000 + ' seconds' +  '</td>');            log('</table>');        }    });</script></html>
Output:
5-Smooth:

1:		1
2:		2
3:		3
4:		4
5:		5
6:		6
7:		8
8:		9
9:		10
10:		12
11:		15
12:		16
13:		18
14:		20
15:		24
16:		25
17:		27
18:		30
19:		32
20:		36
1691:		2125764000
1000000:	519312780448388736089589843750000000000000000000000000000000000000000000000000000000
Elapsed time:	1.73 seconds

7-Smooth:

1:		1
2:		2
3:		3
4:		4
5:		5
6:		6
7:		7
8:		8
9:		9
10:		10
11:		12
12:		14
13:		15
14:		16
15:		18
16:		20
17:		21
18:		24
19:		25
20:		27
1691:		3317760
1000000:	4157409948433216829957008507500000000
Elapsed time:	1.989 seconds

## Liberty BASIC

LB has unlimited precision integers.

 dim h( 1000000) for i =1 to 20    print hamming( i); " ";next i printprint "H( 1691)", hamming( 1691)print "H( 1000000)", hamming( 1000000) end function hamming( limit)    h( 0) =1    x2 =2: x3 =3: x5 =5    i  =0: j  =0: k  =0    for n =1 to limit        h( n) = min( x2, min( x3, x5))        if x2 = h( n) then i = i +1: x2 =2 *h( i)        if x3 = h( n) then j = j +1: x3 =3 *h( j)        if x5 = h( n) then k = k +1: x5 =5 *h( k)    next n    hamming =h( limit -1)end function
1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36
H( 1691)
2125764000
H( 1000000) 519312780448388736089589843750000000000000000000000000000000000000000000000000000000


## Logo

to init.ham  ; queues  make "twos   [1]  make "threes [1]  make "fives  [1]endto next.ham                          localmake "ham first :twos  if less? first :threes :ham [make "ham first :threes]  if less? first :fives  :ham [make "ham first :fives]   if equal? :ham first :twos   [ignore dequeue "twos]  if equal? :ham first :threes [ignore dequeue "threes]  if equal? :ham first :fives  [ignore dequeue "fives]   queue "twos   :ham * 2  queue "threes :ham * 3  queue "fives  :ham * 5   output :hamend init.hamrepeat      20 [print  next.ham]repeat 1690-20 [ignore next.ham]print next.ham

## Lua

function hiter()  hammings = {1}  prev, vals = {1, 1, 1}  index = 1  local function nextv()    local n, v = 1, hammings[prev[1]]*2	if hammings[prev[2]]*3 < v then n, v = 2, hammings[prev[2]]*3 end	if hammings[prev[3]]*5 < v then n, v = 3, hammings[prev[3]]*5 end	prev[n] = prev[n] + 1	if hammings[index] == v then return nextv() end	index = index + 1	hammings[index] = v	return v  end  return nextvend j = hiter()for i = 1, 20 do  print(j())endn, l = 0, 0while n < 2^31 do n, l = j(), n endprint(l)

## Mathematica

HammingList[N_] := Module[{A, B, C}, {A, B, C} = (N^(1/3))*{2.8054745679851933, 1.7700573778298891, 1.2082521307023026} - {1, 1, 1}; Take[ Sort@Flatten@Table[ 2^x * 3^y * 5^z , {x, 0, A}, {y, 0, (-B/A)*x + B}, {z, 0, C - (C/A)*x - (C/B)*y}], N]];
HammingList[20]
-> {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36}
HammingList[1691] // Last
-> 2125764000
HammingList[1000000] // Last
->519312780448388736089589843750000000000000000000000000000000000000000000000000000000

## MUMPS

Hamming(n)	New count,ok,next,number,which	For which=2,3,5 Set number=1	For count=1:1:n Do	. Set ok=0 Set:count<21 ok=1 Set:count=1691 ok=1 Set:count=n ok=1	. Write:ok !,$Justify(count,5),": ",number . For which=2,3,5 Set next(number*which)=which . Set number=$Order(next(""))	. Kill next(number)	. Quit	QuitDo Hamming(2000)     1: 1    2: 2    3: 3    4: 4    5: 5    6: 6    7: 8    8: 9    9: 10   10: 12   11: 15   12: 16   13: 18   14: 20   15: 24   16: 25   17: 27   18: 30   19: 32   20: 36 1691: 2125764000 2000: 8062156800

declare  fun lazy {HammingFun}     1|{FoldL1 [{MultHamming 2} {MultHamming 3} {MultHamming 5}] LMerge}  end   Hamming = {HammingFun}   fun {MultHamming N}     {LMap Hamming fun {$X} N*X end} end fun lazy {LMap Xs F} case Xs of nil then nil [] X|Xr then {F X}|{LMap Xr F} end end fun lazy {LMerge Xs=X|Xr Ys=Y|Yr} if X < Y then X|{LMerge Xr Ys} elseif X > Y then Y|{LMerge Xs Yr} else X|{LMerge Xr Yr} end end fun {FoldL1 X|Xr F} {FoldL Xr F X} endin {ForAll {List.take Hamming 20} System.showInfo} {System.showInfo {Nth Hamming 1690}} {System.showInfo {Nth Hamming 1000000}} ## PARI/GP This is a basic implementation; finding the millionth term requires 3 seconds and 54 MB. Much better algorithms exist. Hupto(n)={ my(v=vector(n),x2=2,x3=3,x5=5,i=1,j=1,k=1,t); v[1]=1; for(m=2,n, v[m]=t=min(x2,min(x3,x5)); if(x2 == t, x2 = v[i++] << 1); if(x3 == t, x3 = 3 * v[j++]); if(x5 == t, x5 = 5 * v[k++]); ); v};H(n)=Hupto(n)[n]; Hupto(20)H(1691)H(10^6) Output: %1 = [1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36] %2 = 2125764000 %3 = 519312780448388736089589843750000000000000000000000000000000000000000000000000000000 ## Perl use List::Util 'min'; sub ham_gen { my @s = ([1], [1], [1]); my @m = (2, 3, 5); return sub { # use bigint; my$n = min($s[0][0],$s[1][0], $s[2][0]); for (0 .. 2) { shift @{$s[$_]} if$s[$_][0] ==$n;			push @{$s[$_]}, $n *$m[$_] } return$n	}} my ($h,$i) = ham_gen; ++$i, print$h->(), " " until $i > 20;print "...\n"; ++$i, $h->() until$i == 1690;print ++$i, "-th: ",$h->(), "\n";
Output:
1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36 40 ...
1691-th: 2125764000
<perl's bigint is so horribly slow that I didn't have the patience for the 1000000th output>
<some big number presumably>

## Perl 6

Works with: Rakudo version 2012.01

The limit scaling is not required, but it cuts down on a bunch of unnecessary calculation.

my $limit = 32; sub powers_of ($radix) { 1, [\*] $radix xx * } my @hammings = ( powers_of(2)[^$limit ]       X*    ( powers_of(3)[^($limit * 2/3)] X* powers_of(5)[^($limit * 1/2)]    )  ).sort; say ~@hammings[^20];say @hammings[1690]; # zero indexed
Output:
1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36
2125764000


## PicoLisp

(de hamming (N)   (let (L (1)  H)      (do N         (for (X L X (cadr X))      # Find smallest result            (setq H (car X)) )         (idx 'L H NIL)             # Remove it         (for I (2 3 5)             # Generate next results            (idx 'L (* I H) T) ) )      H ) ) (println (make (for N 20 (link (hamming N)))))(println (hamming 1691))(println (hamming 1000000))
Output:
(1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36)
2125764000
519312780448388736089589843750000000000000000000000000000000000000000000000000000000

## PL/I

(subscriptrange):Hamming: procedure options (main); /* 14 November 2013 */   declare (H(3000), t) fixed (15);   declare (i, j, k, m, n) fixed binary;   declare swaps bit (1);    on underflow ;    m = 0; n = 12;   do k = 0 to n;      do j = 0 to n;         do i = 0 to n;            m = m + 1;            H(m) = 2**i * 3**j * 5**k;         end;      end;   end;   /* sort */   swaps = '1'b;   do while (swaps); /* Cocktail-shaker sort is adequate, because values are largely sorted */      swaps = '0'b;      do i = 1 to m-1, i-1 to 1 by -1;         if H(i) > H(i+1) then /* swap */            do; t = H(i); H(i) = H(i+1); H(i+1) = t; swaps = '1'b; end;      end;   end;   do i = 1 to m;      put skip data (H(i));   end;   put skip data (H(1653));end Hamming;

Results:

H(1)=                 1;
H(2)=                 2;
H(3)=                 3;
H(4)=                 4;
H(5)=                 5;
H(6)=                 6;
H(7)=                 8;
H(8)=                 9;
H(9)=                10;
H(10)=                12;
H(11)=                15;
H(12)=                16;
H(13)=                18;
H(14)=                20;
H(15)=                24;
H(16)=                25;
H(17)=                27;
H(18)=                30;
H(19)=                32;
H(20)=                36;

## Prolog

### Generator idiom

%% collect N elements produced by a generator in a row take( 0, Next, Z-Z, Next).take( N, Next, [A|B]-Z, NZ):- N>0, !, next(Next,A,Next1),  N1 is N-1,  take(N1,Next1,B-Z,NZ). %% a generator provides specific {next} implementation next( hamm( A2,B,C3,D,E5,F,[H|G] ), H, hamm(X,U,Y,V,Z,W,G) ):-   H is min(A2, min(C3,E5)),  (   A2 =:= H -> B=[N2|U],X is N2*2 ; (X,U)=(A2,B) ),  (   C3 =:= H -> D=[N3|V],Y is N3*3 ; (Y,V)=(C3,D) ),  (   E5 =:= H -> F=[N5|W],Z is N5*5 ; (Z,W)=(E5,F) ). mkHamm( hamm(1,X,1,X,1,X,X) ).       % Hamming numbers generator init state main(N) :-     mkHamm(G),take(20,G,A-[],_),           write(A), nl,     take(1691-1,G,_,G2),take(2,G2,B-[],_),     write(B), nl,      take(  N  -1,G,_,G3),take(2,G3,[C1|_]-_,_),   write(C1), nl.

SWI Prolog 6.2.6 produces (in about 7 ideone seconds):

 ?- time( main(1000000) ).
[1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,27,30,32,36]
[2125764000,2147483648]
519312780448388736089589843750000000000000000000000000000000000000000000000000000000
% 10,017,142 inferences


### Laziness flavor

Works with SWI-Prolog. Laziness is simulate with freeze/2 and ground/2.
Took inspiration from this code : http://chr.informatik.uni-ulm.de/~webchr (click on hamming.pl: Solves Hamming Problem).

hamming(N) :-     % to stop cleanly     nb_setval(go, 1),      % display list     (	 N = 20 -> watch_20(20, L); watch(1,N,L)),      % go     L=[1|L235],     multlist(L,2,L2),     multlist(L,3,L3),     multlist(L,5,L5),     merge_(L2,L3,L23),     merge_(L5,L23,L235).  %% multlist(L,N,LN)%% multiply each element of list L with N, resulting in list LN%% here only do multiplication for 1st element, then use multlist recursivelymultlist([X|L],N,XLN) :-	% the trick to stop	nb_getval(go, 1) -> 	% laziness flavor		when(ground(X),	     (	 XN is X*N,		 XLN=[XN|LN],		 multlist(L,N,LN))); 	true. merge_([X|In1],[Y|In2],XYOut) :-	% the trick to stop	nb_getval(go, 1) -> 	% laziness flavor	(   X < Y -> XYOut = [X|Out], In11 = In1, In12 = [Y|In2]	;   X = Y -> XYOut = [X|Out], In11 = In1, In12 = In2	;            XYOut = [Y|Out], In11 = [X | In1], In12 = In2),	freeze(In11,freeze(In12, merge_(In11,In12,Out))); 	true. %% display nth elementwatch(Max, Max, [X|_]) :-	% laziness flavor	when(ground(X),	     (format('~w~n', [X]), 	      % the trick to stop	      nb_linkval(go, 0))).  watch(N, Max, [_X|L]):-	 N1 is N + 1,	 watch(N1, Max, L).  %% display nth elementwatch_20(1, [X|_]) :-	% laziness flavor	when(ground(X),	     (format('~w~n', [X]), 	      % the trick to stop	      nb_linkval(go, 0))).  watch_20(N, [X|L]):-	% laziness flavor	when(ground(X),	     (format('~w ', [X]),	      N1 is N - 1,	      watch_20(N1, L))).
Output:
?- hamming(20).
1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36
true .

?- hamming(1691).
2125764000
true .

?- hamming(1000000).
519312780448388736089589843750000000000000000000000000000000000000000000000000000000
true .


## Python

### Version based on example from Dr. Dobb's CodeTalk

from itertools import islice def hamming2():    '''\    This version is based on a snippet from:        http://dobbscodetalk.com/index.php?option=com_content&task=view&id=913&Itemid=85         When expressed in some imaginary pseudo-C with automatic        unlimited storage allocation and BIGNUM arithmetics, it can be        expressed as:            hamming = h where              array h;              n=0; h[0]=1; i=0; j=0; k=0;              x2=2*h[ i ]; x3=3*h[j]; x5=5*h[k];              repeat:                h[++n] = min(x2,x3,x5);                if (x2==h[n]) { x2=2*h[++i]; }                if (x3==h[n]) { x3=3*h[++j]; }                if (x5==h[n]) { x5=5*h[++k]; }     '''    h = 1    _h=[h]    # memoized    multipliers  = (2, 3, 5)    multindeces  = [0 for i in multipliers] # index into _h for multipliers    multvalues   = [x * _h[i] for x,i in zip(multipliers, multindeces)]    yield h    while True:        h = min(multvalues)        _h.append(h)        for (n,(v,x,i)) in enumerate(zip(multvalues, multipliers, multindeces)):            if v == h:                i += 1                multindeces[n] = i                multvalues[n]  = x * _h[i]        # cap the memoization        mini = min(multindeces)        if mini >= 1000:            del _h[:mini]            multindeces = [i - mini for i in multindeces]        #        yield h
Output:
>>> list(islice(hamming2(), 20))
[1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36]
>>> list(islice(hamming2(), 1690, 1691))
[2125764000]
>>> list(islice(hamming2(), 999999, 1000000))
[519312780448388736089589843750000000000000000000000000000000000000000000000000000000]


### Another implementation of same approach

This version uses a lot of memory, it doesn't try to limit memory usage.

import psyco def hamming(limit):    h = [1] * limit    x2, x3, x5 = 2, 3, 5    i = j = k = 0     for n in xrange(1, limit):        h[n] = min(x2, x3, x5)        if x2 == h[n]:            i += 1            x2 = 2 * h[i]        if x3 == h[n]:            j += 1            x3 = 3 * h[j]        if x5 == h[n]:            k += 1            x5 = 5 * h[k]     return h[-1] psyco.bind(hamming)print [hamming(i) for i in xrange(1, 21)]print hamming(1691)print hamming(1000000)

### "Cyclical Iterators"

The original author is Raymond Hettinger and the code was first published here under the MIT license. Uses iterators dubbed "cyclical" in a sense that they are referring back (explicitly, with p2, p3, p5 iterators) to the previously produced values, same as the above versions (through indecies into shared storage) and the classic Haskell version (implicitly timed by lazy evaluation).

Memory is efficiently maintained automatically by the tee function for each of the three generator expressions, i.e. only that much is maintained as needed to produce the next value (although it looks like the storage is not shared so three copies are maintained implicitly there).

from itertools import tee, chain, groupby, islicefrom heapq import merge def raymonds_hamming():    # Generate "5-smooth" numbers, also called "Hamming numbers"    # or "Regular numbers".  See: http://en.wikipedia.org/wiki/Regular_number    # Finds solutions to 2**i * 3**j * 5**k  for some integers i, j, and k.     def deferred_output():        for i in output:            yield i     result, p2, p3, p5 = tee(deferred_output(), 4)    m2 = (2*x for x in p2)                          # multiples of 2    m3 = (3*x for x in p3)                          # multiples of 3    m5 = (5*x for x in p5)                          # multiples of 5    merged = merge(m2, m3, m5)    combined = chain([1], merged)                   # prepend a starting point    output = (k for k,g in groupby(combined))       # eliminate duplicates     return result print list(islice(raymonds_hamming(), 20))print islice(raymonds_hamming(), 1689, 1690).next()print islice(raymonds_hamming(), 999999, 1000000).next()

Results are the same as before.

#### Non-sharing recursive generator

Another formulation along the same lines, but greatly simplified, found here. Lacks data sharing, i.e. calls self recursively thus creating a separate copy of the data stream fed to the tee() call, again and again, instead of using its own output. This gravely impacts the efficiency. Not to be used.

from heapq import mergefrom itertools import tee def hamming_numbers():    last = 1    yield last     a,b,c = tee(hamming_numbers(), 3)     for n in merge((2*i for i in a), (3*i for i in b), (5*i for i in c)):        if n != last:            yield n            last = n

#### Cyclic generator method #2.

Cyclic generator method #2. Considerably faster due to early elimination (before merge) of duplicates. Currently the faster Python version. Direct copy of Haskell code.

from itertools import islice, chain, tee def merge(r, s):    # This is faster than heapq.merge.    rr = r.next()    ss = s.next()    while True:        if rr < ss:            yield rr            rr = r.next()        else:            yield ss            ss = s.next() def p(n):    def gen():        x = n        while True:            yield x            x *= n    return gen() def pp(n, s):    def gen():        for x in (merge(s, chain([n], (n * y for y in fb)))):            yield x    r, fb = tee(gen())    return r def hamming(a, b = None):    if not b:        b = a + 1    seq = (chain([1], pp(5, pp(3, p(2)))))    return list(islice(seq, a - 1, b - 1)) print hamming(1, 21)print hamming(1691)[0]print hamming(1000000)[0]

## Qi

 This example is incomplete. Parts 2 & 3 of task missing. Please ensure that it meets all task requirements and remove this message.
Translation of: Clojure
(define smerge  [X|Xs] [Y|Ys] -> [X | (freeze (smerge (thaw Xs) [Y|Ys]))] where (< X Y)  [X|Xs] [Y|Ys] -> [Y | (freeze (smerge [X|Xs] (thaw Ys)))] where (> X Y)  [X|Xs] [_|Ys] -> [X | (freeze (smerge (thaw Xs) (thaw Ys)))]) (define smerge3  Xs Ys Zs -> (smerge Xs (smerge Ys Zs))) (define smap  F [S|Ss] -> [(F S)|(freeze (smap F (thaw Ss)))]) (set hamming [1 | (freeze (smerge3 (smap (* 2) (value hamming))                                   (smap (* 3) (value hamming))                                   (smap (* 5) (value hamming))))]) (define stake  _      0 -> []  [S|Ss] N -> [S|(stake (thaw Ss) (1- N))]) (stake (value hamming) 20)
Output:
[1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36]


## R

Recursively find the Hamming numbers below 231. Works for larger limits, however to find the 1000000th value, one needs guess the correct limit

hamming=function(hamms,limit) {  tmp=hamms  for(h in c(2,3,5)) {    tmp=c(tmp,h*hamms)  }  tmp=unique(tmp[tmp<=limit])  if(length(tmp)>length(hamms)) {    hamms=hamming(tmp,limit)  }  hamms}sort(hamming(1,limit=2^31)[-1])

## Racket

 #lang racket(require racket/stream)(define first stream-first)(define rest  stream-rest) (define (merge s1 s2)  (define x1 (first s1))  (define x2 (first s2))  (cond [(= x1 x2) (merge s1 (rest s2))]        [(< x1 x2) (stream-cons x1 (merge (rest s1) s2))]        [else      (stream-cons x2 (merge s1 (rest s2)))])) (define (mult k) (λ(x) (* x k))) (define hamming  (stream-cons    1 (merge (stream-map (mult 2) hamming)            (merge (stream-map (mult 3) hamming)                   (stream-map (mult 5) hamming))))) (for/list ([i 20] [x hamming]) x)(stream-ref hamming 1690)(stream-ref hamming 999999) 

Output:

'(1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36)
2125764000
519312780448388736089589843750000000000000000000000000000000000000000000000000000000

## Raven

Translation of: Liberty Basic
define hamming use $limit [ ] as$h    1 $h 0 set 2 as$x2   3 as $x3 5 as$x5    0 as $i 0 as$j     0 as $k 1$limit 1 + 1 range each as $n$x3 $x5 min$x2 min    $h$n   set        $h$n get   $x2 = if$i  1 +   as $i$h $i get 2 * as$x2        $h$n get   $x3 = if$j  1 +   as $j$h $j get 3 * as$x3        $h$n get   $x5 = if$k  1 +   as $k$h $k get 5 * as$x5    $h$limit 1 -   get 1 21 1 range each as $lim$lim hamming print " " print"\n" print "Hamming(1691) is: " print    1691 hamming print    "\n" print # Raven can't handle > 2^31 using integers##"Hamming(1000000) is: " print   1000000 hamming print    "\n" print
Output:
1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36
Hamming(1691) is: 2125764000

## REXX

### concise version

This REXX program was a direct translatation from my old REXX subroutine to compute UGLY numbers.
The program computes enough Hamming numbers (just one past the current number).

/*REXX program computes Hamming numbers: 1──►20,  #1691,  one millionth.*/numeric digits 100                     /*ensure we have enough precision*/call hamming 1, 20                     /*show the first ──► twentieth #s*/call hamming 1691                      /*show the 1,691st Hamming number*/call hamming 1000000                   /*show the  one millionth number.*/exit                                   /*stick a fork in it, we're done.*//*──────────────────────────────────HAMMING subroutine──────────────────*/hamming:  procedure;   parse arg x,y;   if y=='' then y=x;     w=length(y)          #2=1;     #3=1;     #5=1;     @.=0;    @.1=1    do n=2  for y-1   @.n = min(2*@.#2, 3*@.#3, 5*@.#5)   /*pick the minimum of three pigs.*/   if 2*@.#2 == @.n   then #2 = #2+1   /*# already defined?  Use next #.*/   if 3*@.#3 == @.n   then #3 = #3+1   /*"    "       "       "    "  " */   if 5*@.#5 == @.n   then #5 = #5+1   /*"    "       "       "    "  " */   end   /*n*/                  do j=x  to y         /*W  is used to align the index. */                  say 'Hamming('right(j,w)") ="  @.j   /*list 'em, Dano.*/                  end   /*j*/ say right( 'length of last Hamming number ='  length(@.y),  70);       sayreturn

output when using the default input(s):

Hamming( 1) = 1
Hamming( 2) = 2
Hamming( 3) = 3
Hamming( 4) = 4
Hamming( 5) = 5
Hamming( 6) = 6
Hamming( 7) = 8
Hamming( 8) = 9
Hamming( 9) = 10
Hamming(10) = 12
Hamming(11) = 15
Hamming(12) = 16
Hamming(13) = 18
Hamming(14) = 20
Hamming(15) = 24
Hamming(16) = 25
Hamming(17) = 27
Hamming(18) = 30
Hamming(19) = 32
Hamming(20) = 36
length of last Hamming number = 2

Hamming(1691) = 2125764000
length of last Hamming number = 10

Hamming(1000000) = 519312780448388736089589843750000000000000000000000000000000000000000000000000000000
length of last Hamming number = 84


### unrolled version

This REXX version is roughly twice as fast as the first version.

/*REXX program computes Hamming numbers: 1──►20,  #1691,  one millionth.*/numeric digits 100                     /*ensure we have enough precision*/call hamming 1, 20                     /*show the first ──► twentieth #s*/call hamming 1691                      /*show the 1,691st Hamming number*/call hamming 1000000                   /*show the  one millionth number.*/exit                                   /*stick a fork in it, we're done.*//*──────────────────────────────────HAMMING subroutine──────────────────*/hamming:  procedure;   parse arg x,y;   if y=='' then y=x;     w=length(y)          #2=1;     #3=1;     #5=1;     @.=0;    @.1=1    do n=2  for y-1   _2 = @.#2 + @.#2                    /*this is faster than   2 * @.#2 */   _3 =    3 * @.#3   _5 =    5 * @.#5                     m =_2             /*assume a minimum (of the three)*/   if _3  < m   then m =_3             /*is this less than the minimum? */   if _5  < m   then m =_5             /* "   "    "    "   "     "     */      @.n = m                          /*now, assign the next Hamming #.*/   if _2 == m   then #2 = #2 + 1       /*# already defined?  Use next #.*/   if _3 == m   then #3 = #3 + 1       /*"    "       "       "    "  " */   if _5 == m   then #5 = #5 + 1       /*"    "       "       "    "  " */   end   /*n*/                  do j=x  to y         /*W  is used to align the index. */                  say 'Hamming('right(j,w)") ="  @.j   /*list 'em, Dano.*/                  end   /*j*/ say right( 'length of last Hamming number ='  length(@.y),  70);       sayreturn

output is identical to the first version.

## Ruby

Translation of: Scala
Works with: Ruby version 1.8.6
require 'generator' # the Hamming number generatorhamming = Generator.new do |generator|  next_ham = 1  queues = { 2 => [], 3 => [], 5 => [] }  loop do    generator.yield next_ham     [2,3,5].each {|m| queues[m] << (next_ham * m)}    next_ham = [2,3,5].collect {|m| queues[m][0]}.min    [2,3,5].each {|m| queues[m].shift if queues[m][0] == next_ham}  endend
Works with: Ruby version 1.9.1

This method does not require a library module.

hamming = Enumerator.new do |yielder|  next_ham = 1  queues = { 2 => [], 3 => [], 5 => [] }   loop do    yielder << next_ham   # or: yielder.yield(next_ham)     [2,3,5].each {|m| queues[m]<< (next_ham * m)}    next_ham = [2,3,5].collect {|m| queues[m][0]}.min    [2,3,5].each {|m| queues[m].shift if queues[m][0]== next_ham}  endend

And the "main" part of the task

start = Time.now idx = 1hamming.each do |ham|  case idx  when (1..20), 1691    p [idx, ham]  when 1_000_000    p [idx, ham]    break  end  idx += 1end puts "elapsed: #{Time.now - start} seconds"
Output:
[1, 1]
[2, 2]
[3, 3]
[4, 4]
[5, 5]
[6, 6]
[7, 8]
[8, 9]
[9, 10]
[10, 12]
[11, 15]
[12, 16]
[13, 18]
[14, 20]
[15, 24]
[16, 25]
[17, 27]
[18, 30]
[19, 32]
[20, 36]
[1691, 2125764000]
[1000000, 519312780448388736089589843750000000000000000000000000000000000000000000000000000000]
elapsed: 143.96875 seconds

## Run BASIC

 dim h(1000000)for i =1 to 20    print hamming(i);" ";next i printprint "Hamming List First(1691)   =";chr$(9);hamming(1691)print "Hamming List Last(1000000) =";chr$(9);hamming(1000000) end function hamming(limit)    h(0) =1    x2 = 2: x3 = 3: x5 =5    i  = 0: j  = 0: k  =0    for n =1 to limit        h(n)  = min(x2, min(x3, x5))        if x2 = h(n) then i = i +1: x2 =2 *h(i)        if x3 = h(n) then j = j +1: x3 =3 *h(j)        if x5 = h(n) then k = k +1: x5 =5 *h(k)    next n    hamming = h(limit -1)end function
1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36
Hamming List First(1691)   =	2125764000
Hamming List Last(1000000) =	519312780448388736089589843750000000000000000000000000000000000000000000000000000000

## Scala

class Hamming extends Iterator[BigInt] {  import scala.collection.mutable.Queue  val qs = Seq.fill(3)(new Queue[BigInt])  def enqueue(n: BigInt) = qs zip Seq(2, 3, 5) foreach { case (q, m) => q enqueue n * m }  def next = {    val n = qs map (_.head) min;    qs foreach { q => if (q.head == n) q.dequeue }    enqueue(n)    n  }  def hasNext = true  qs foreach (_ enqueue 1)}

However, the usage of closures adds a significant amount of time. The code below, though a bit uglier because of the repetitions, is twice as fast:

class Hamming extends Iterator[BigInt] {  import scala.collection.mutable.Queue  val q2 = new Queue[BigInt]  val q3 = new Queue[BigInt]  val q5 = new Queue[BigInt]  def enqueue(n: BigInt) = {    q2 enqueue n * 2    q3 enqueue n * 3    q5 enqueue n * 5  }  def next = {    val n = q2.head min q3.head min q5.head    if (q2.head == n) q2.dequeue    if (q3.head == n) q3.dequeue    if (q5.head == n) q5.dequeue    enqueue(n)    n  }  def hasNext = true  List(q2, q3, q5) foreach (_ enqueue 1)}

Usage:

scala> new Hamming take 20 toList
res87: List[BigInt] = List(1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36)

scala> new Hamming drop 1690 next
res88: BigInt = 2125764000

scala> new Hamming drop 999999 next
res89: BigInt = 519312780448388736089589843750000000000000000000000000000000000000000000000000000000


There's also a fairly mechanical translation from Haskell using purely functional lazy streams

val hamming : Stream[BigInt] = {   def merge(inx : Stream[BigInt], iny : Stream[BigInt]) : Stream[BigInt] = {      if (inx.head < iny.head) inx.head #:: merge(inx.tail, iny) else       if (iny.head < inx.head) iny.head #:: merge(inx, iny.tail) else      merge(inx, iny.tail)   }    1 #:: merge(hamming map (_ * 2), merge(hamming map (_ * 3), hamming map (_ * 5)))}

Use of "force" ensures that the stream is computed before being printed, otherwise it would just be left suspended and you'd see "Stream(1, ?)"

scala> (hamming take 20).force
res0: scala.collection.immutable.Stream[BigInt] = Stream(1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36)


To get the nth code find the n-1th element because indexes are 0 based

scala> hamming(1690)
res1: BigInt = 2125764000


To calculate the 1000000th code I had to increase the JVM heap from the default

scala> hamming(999999)
res2: BigInt = 519312780448388736089589843750000000000000000000000000000000000000000000000000000000


## Scheme

(define-syntax lons  (syntax-rules ()    ((_ lar ldr) (delay (cons lar (delay ldr)))))) (define (lar lons)  (car (force lons))) (define (ldr lons)  (force (cdr (force lons)))) (define (lap proc . llists)  (lons (apply proc (map lar llists)) (apply lap proc (map ldr llists)))) (define (take n llist)  (if (zero? n)      (list)      (cons (lar llist) (take (- n 1) (ldr llist))))) (define (llist-ref n llist)  (if (= n 1)      (lar llist)      (llist-ref (- n 1) (ldr llist)))) (define (merge llist-1 . llists)  (define (merge-2 llist-1 llist-2)    (cond ((null? llist-1) llist-2)          ((null? llist-2) llist-1)          ((< (lar llist-1) (lar llist-2))           (lons (lar llist-1) (merge-2 (ldr llist-1) llist-2)))          ((> (lar llist-1) (lar llist-2))           (lons (lar llist-2) (merge-2 llist-1 (ldr llist-2))))          (else (lons (lar llist-1) (merge-2 (ldr llist-1) (ldr llist-2))))))  (if (null? llists)      llist-1      (apply merge (cons (merge-2 llist-1 (car llists)) (cdr llists))))) (define hamming  (lons 1        (merge (lap (lambda (x) (* x 2)) hamming)               (lap (lambda (x) (* x 3)) hamming)               (lap (lambda (x) (* x 5)) hamming)))) (display (take 20 hamming))(newline)(display (llist-ref 1691 hamming))(newline)(display (llist-ref 1000000 hamming))(newline)
Output:
(1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36)
2125764000
out of memory

$include "seed7_05.s7i"; include "bigint.s7i"; const func bigInteger: min (in bigInteger: a, in bigInteger: b, in bigInteger: c) is func result var bigInteger: min is 0_; begin if a < b then min := a; else min := b; end if; if c < min then min := c; end if; end func; const func bigInteger: hamming (in integer: n) is func result var bigInteger: hammingNum is 1_; local var array bigInteger: hammingNums is 0 times 0_; var integer: index is 0; var bigInteger: x2 is 2_; var bigInteger: x3 is 3_; var bigInteger: x5 is 5_; var integer: i is 1; var integer: j is 1; var integer: k is 1; begin hammingNums := n times 1_; for index range 2 to n do hammingNum := min(x2, x3, x5); hammingNums[index] := hammingNum; if x2 = hammingNum then incr(i); x2 := 2_ * hammingNums[i]; end if; if x3 = hammingNum then incr(j); x3 := 3_ * hammingNums[j]; end if; if x5 = hammingNum then incr(k); x5 := 5_ * hammingNums[k]; end if; end for; end func; const proc: main is func local var integer: n is 0; begin for n range 1 to 20 do write(hamming(n) <& " "); end for; writeln; writeln(hamming(1691)); writeln(hamming(1000000)); end func; Output: 1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36 2125764000 519312780448388736089589843750000000000000000000000000000000000000000000000000000000  ## Smalltalk Works with: GNU Smalltalk This is a straightforward implementation of the pseudocode snippet found in the Python section. Smalltalk supports arbitrary-precision integers, but the implementation is too slow to try it with 1 million. Object subclass: Hammer [ Hammer class >> hammingNumbers: howMany [ |h i j k x2 x3 x5| h := OrderedCollection new. i := 0. j := 0. k := 0. h add: 1. x2 := 2. x3 := 2. x5 := 5. [ ( h size) < howMany ] whileTrue: [ |m| m := { x2. x3. x5 } sort first. (( h indexOf: m ) = 0) ifTrue: [ h add: m ]. ( x2 = (h last) ) ifTrue: [ i := i + 1. x2 := 2 * (h at: i) ]. ( x3 = (h last) ) ifTrue: [ j := j + 1. x3 := 3 * (h at: j) ]. ( x5 = (h last) ) ifTrue: [ k := k + 1. x5 := 5 * (h at: k) ]. ]. ^ h sort ]]. (Hammer hammingNumbers: 20) displayNl.(Hammer hammingNumbers: 1690) last displayNl. ## Tcl This uses coroutines to simplify the description of what's going on. Works with: Tcl version 8.6 package require Tcl 8.6 # Simple helper: Tcl-style list "map"proc map {varName list script} { set l {} upvar 1$varName v    foreach v $list {lappend l [uplevel 1$script]}    return $l} # The core of a coroutine to compute the product of a hamming sequence.## Tricky bit: we don't automatically advance to the next value, and instead# wait to be told that the value has been consumed (i.e., is the result of# the [yield] operation).proc ham {key multiplier} { global hammingCache set i 0 yield [info coroutine] # Cannot use [foreach]; that would take a snapshot of the list in # the hammingCache variable, so missing updates. while 1 { set n [expr {[lindex$hammingCache($key)$i] * $multiplier}] # If the number selected was ours, we advance to compute the next if {[yield$n] == $n} { incr i } }} # This coroutine computes the hamming sequence given a list of multipliers.# It uses the [ham] helper from above to generate indivdual multiplied# sequences. The key into the cache is the list of multipliers.## Note that it is advisable for the values to be all co-prime wrt each other.proc hammingCore args { global hammingCache set hammingCache($args) 1    set hammers [map x $args {coroutine ham$x,$args ham$args $x}] yield while 1 { set n [lindex$hammingCache($args) [incr i]-1] lappend hammingCache($args) \	    [tcl::mathfunc::min {*}[map h $hammers {$h $n}]] yield$n    }} # Assemble the pieces so as to compute the classic hamming sequence.coroutine hamming hammingCore 2 3 5# Print the first 20 values of the sequencefor {set i 1} {$i <= 20} {incr i} { puts [format "hamming$%d$ = %d"$i [hamming]]}for {} {$i <= 1690} {incr i} {set h [hamming]}puts "hamming{1690} =$h"for {} {$i <= 1000000} {incr i} {set h [hamming]}puts "hamming{1000000} =$h"
Output:
hamming{1} = 1
hamming{2} = 2
hamming{3} = 3
hamming{4} = 4
hamming{5} = 5
hamming{6} = 6
hamming{7} = 8
hamming{8} = 9
hamming{9} = 10
hamming{10} = 12
hamming{11} = 15
hamming{12} = 16
hamming{13} = 18
hamming{14} = 20
hamming{15} = 24
hamming{16} = 25
hamming{17} = 27
hamming{18} = 30
hamming{19} = 32
hamming{20} = 36
hamming{1690} = 2123366400
hamming{1000000} = 519312780448388736089589843750000000000000000000000000000000000000000000000000000000


A faster version can be built that also works on Tcl 8.5 (or earlier, if only small hamming numbers are being computed):

variable hamming 1 hi2 0 hi3 0 hi5 0proc hamming {n} {    global hamming hi2 hi3 hi5    set h2 [expr {[lindex $hamming$hi2]*2}]    set h3 [expr {[lindex $hamming$hi3]*3}]    set h5 [expr {[lindex $hamming$hi5]*5}]    while {[llength $hamming] <$n} {	lappend hamming [set h [expr {	    $h2<$h3	        ? $h2<$h5 ? $h2 :$h5	        : $h3<$h5 ? $h3 :$h5	}]]	if {$h==$h2} {	    set h2 [expr {[lindex $hamming [incr hi2]]*2}] } if {$h==$h3} { set h3 [expr {[lindex$hamming [incr hi3]]*3}]	}	if {$h==$h5} {	    set h5 [expr {[lindex $hamming [incr hi5]]*5}] } } return [lindex$hamming [expr {$n - 1}]]} # Print the first 20 values of the sequencefor {set i 1} {$i <= 20} {incr i} {    puts [format "hamming$%d$ = %d" $i [hamming$i]]}puts "hamming{1690} = [hamming 1690]"puts "hamming{1691} = [hamming 1691]"puts "hamming{1692} = [hamming 1692]"puts "hamming{1693} = [hamming 1693]"puts "hamming{1000000} = [hamming 1000000]"

## Ursala

Smooth is defined as a second order function taking a list of primes and returning a function that takes a natural number n to the n-th smooth number with respect to them. An elegant but inefficient formulation based on the J solution is the following.

#import std#import nat smooth"p" "n" = ~&z take/"n" nleq-< (rep(length "n") ^Ts/~& product*K0/"p") <1>

This test program

main = smooth<2,3,5>* nrange(1,20)

yields this list of the first 20 Hamming numbers.

<1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,27,30,32,36>


Although all calculations are performed using unlimited precision, the version above is impractical for large numbers. A more hardcore approach is the following.

#import std#import nat smooth"p" "n" = ~&H\"p" *-<1>; @NiXS ~&/(1,1); ~&ll~="n"->lr -+   ^\~&rlPrrn2rrm2Zlrrmz3EZYrrm2lNCTrrm2QAX*rhlPNhrnmtPA2XtCD ~&lrPrhl2E?/~&l ^|/successor@l ~&hl,   ^|/~& nleq-<&l+ * ^\~&r ~&l|| product@rnmhPX+- #cast %nL main = smooth<2,3,5>* nrange(1,20)--<1691,1000000>
Output:

The great majority of time is spent calculating the millionth Hamming number.

<
1,
2,
3,
4,
5,
6,
8,
9,
10,
12,
15,
16,
18,
20,
24,
25,
27,
30,
32,
36,
2125764000,
519312780448388736089589843750000000000000000000000000000000000000000000000000000000>`