N-smooth numbers

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

n-smooth   numbers are positive integers which have no prime factors > n.

The   n   (when using it in the expression)   n-smooth   is always prime,
there are   no   9-smooth numbers.

1   (unity)   is always included in n-smooth numbers.



2-smooth   numbers are non-negative powers of two.
5-smooth   numbers are also called   Hamming numbers.
7-smooth   numbers are also called    humble   numbers.


A way to express   11-smooth   numbers is:

  11-smooth  =  2i × 3j × 5k × 7m × 11p
           where     i, j, k, m, p ≥ 0  


Task
  •   calculate and show the first   25   n-smooth numbers   for   n=2   ───►   n=29
  •   calculate and show   three numbers starting with   3,000   n-smooth numbers   for   n=3   ───►   n=29
  •   calculate and show twenty numbers starting with  30,000   n-smooth numbers   for   n=503   ───►   n=521   (optional)


All ranges   (for   n)   are to be inclusive, and only prime numbers are to be used.
The (optional) n-smooth numbers for the third range are:   503,   509,   and   521.
Show all n-smooth numbers for any particular   n   in a horizontal list.
Show all output here on this page.


Related tasks


References



11l[edit]

Translation of: Python
V primes = [2, 3, 5, 7, 11, 13, 17, 19, 23]

F isPrime(n)
   I n < 2
      R 0B

   L(i) :primes
      I n == i
         R 1B
      I n % i == 0
         R 0B
      I i * i > n
         R 1B
   print(‘Oops, ’n‘  is too large’)
   R 0B

F init()
   V s = 24
   L s < 600
      I isPrime(s - 1) & s - 1 > :primes.last
         :primes.append(s - 1)
      I isPrime(s + 1) & s + 1 > :primes.last
         :primes.append(s + 1)
      s += 6

F nsmooth(n, size)
   assert(n C 2..521)
   assert(size >= 1)

   V bn = n
   V ok = 0B
   L(prime) :primes
      I bn == prime
         ok = 1B
         L.break
   assert(ok, ‘must be a prime number’)

   V ns = [BigInt(0)] * size
   ns[0] = 1

   [BigInt] next
   L(prime) :primes
      I prime > bn
         L.break
      next.append(prime)

   V indicies = [0] * next.len
   L(m) 1 .< size
      ns[m] = min(next)
      L(i) 0 .< indicies.len
         I ns[m] == next[i]
            indicies[i]++
            next[i] = :primes[i] * ns[indicies[i]]

   R ns

init()

L(p) primes
   I p >= 30
      L.break
   print(‘The first ’p‘ -smooth numbers are:’)
   print(nsmooth(p, 25))
   print()

L(p) primes[1..]
   I p >= 30
      L.break
   print(‘The 3000 to 3202 ’p‘ -smooth numbers are:’)
   print(nsmooth(p, 3002)[2999..])
   print()

L(p) [503, 509, 521]
   print(‘The 30000 to 3019 ’p‘ -smooth numbers are:’)
   print(nsmooth(p, 30019)[29999..])
   print()
Output:
The first 2 -smooth numbers are:
[1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216]

The first 3 -smooth numbers are:
[1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192]

The first 5 -smooth numbers are:
[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]

The first 7 -smooth numbers are:
[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]

The first 11 -smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32]

The first 13 -smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28]

The first 17 -smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27]

The first 19 -smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26]

The first 23 -smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]

The first 29 -smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]

The 3000 to 3202 3 -smooth numbers are:
[91580367978306252441724649472, 92829823186414819915547541504, 94096325042746502515294076928]

The 3000 to 3202 5 -smooth numbers are:
[278942752080, 279936000000, 281250000000]

The 3000 to 3202 7 -smooth numbers are:
[50176000, 50331648, 50388480]

The 3000 to 3202 11 -smooth numbers are:
[2112880, 2116800, 2117016]

The 3000 to 3202 13 -smooth numbers are:
[390000, 390390, 390625]

The 3000 to 3202 17 -smooth numbers are:
[145800, 145860, 146016]

The 3000 to 3202 19 -smooth numbers are:
[74256, 74358, 74360]

The 3000 to 3202 23 -smooth numbers are:
[46552, 46575, 46585]

The 3000 to 3202 29 -smooth numbers are:
[33516, 33524, 33534]

The 30000 to 3019 503 -smooth numbers are:
[62913, 62914, 62916, 62918, 62920, 62923, 62926, 62928, 62930, 62933, 62935, 62937, 62944, 62946, 62951, 62952, 62953, 62957, 62959, 62964]

The 30000 to 3019 509 -smooth numbers are:
[62601, 62602, 62604, 62607, 62608, 62609, 62611, 62618, 62620, 62622, 62624, 62625, 62626, 62628, 62629, 62634, 62640, 62643, 62645, 62646]

The 30000 to 3019 521 -smooth numbers are:
[62287, 62288, 62291, 62292, 62300, 62304, 62307, 62308, 62310, 62315, 62320, 62321, 62322, 62325, 62328, 62329, 62330, 62331, 62335, 62336]

C[edit]

Library: GMP
#include <stdbool.h>
#include <stdint.h>
#include <stdio.h>
#include <stdlib.h>
#include <gmp.h>

void* xmalloc(size_t n) {
    void* ptr = malloc(n);
    if (ptr == NULL) {
        fprintf(stderr, "Out of memory\n");
        exit(1);
    }
    return ptr;
}

void* xrealloc(void* p, size_t n) {
    void* ptr = realloc(p, n);
    if (ptr == NULL) {
        fprintf(stderr, "Out of memory\n");
        exit(1);
    }
    return ptr;
}

bool is_prime(uint32_t n) {
    if (n == 2)
        return true;
    if (n < 2 || n % 2 == 0)
        return false;
    for (uint32_t p = 3; p * p <= n; p += 2) {
        if (n % p == 0)
            return false;
    }
    return true;
}

// Populates primes with the prime numbers between from and to and
// returns the number of primes found.
uint32_t find_primes(uint32_t from, uint32_t to, uint32_t** primes) {
    uint32_t count = 0, buffer_length = 16;
    uint32_t* buffer = xmalloc(sizeof(uint32_t) * buffer_length);
    for (uint32_t p = from; p <= to; ++p) {
        if (is_prime(p)) {
            if (count >= buffer_length) {
                uint32_t new_length = buffer_length * 2;
                if (new_length < count + 1)
                    new_length = count + 1;
                buffer = xrealloc(buffer, sizeof(uint32_t) * new_length);
                buffer_length = new_length;
            }
            buffer[count++] = p;
        }
    }
    *primes = buffer;
    return count;
}

void free_numbers(mpz_t* numbers, size_t count) {
    for (size_t i = 0; i < count; ++i)
        mpz_clear(numbers[i]);
    free(numbers);
}

// Returns an array containing first count n-smooth numbers
mpz_t* find_nsmooth_numbers(uint32_t n, uint32_t count) {
    uint32_t* primes = NULL;
    uint32_t num_primes = find_primes(2, n, &primes);
    mpz_t* numbers = xmalloc(sizeof(mpz_t) * count);
    mpz_t* queue = xmalloc(sizeof(mpz_t) * num_primes);
    uint32_t* index = xmalloc(sizeof(uint32_t) * num_primes);
    for (uint32_t i = 0; i < num_primes; ++i) {
        index[i] = 0;
        mpz_init_set_ui(queue[i], primes[i]);
    }
    for (uint32_t i = 0; i < count; ++i)
        mpz_init(numbers[i]);
    mpz_set_ui(numbers[0], 1);
    for (uint32_t i = 1; i < count; ++i) {
        for (uint32_t p = 0; p < num_primes; ++p) {
            if (mpz_cmp(queue[p], numbers[i - 1]) == 0)
                mpz_mul_ui(queue[p], numbers[++index[p]], primes[p]);
        }
        uint32_t min_index = 0;
        for (uint32_t p = 1; p < num_primes; ++p) {
            if (mpz_cmp(queue[min_index], queue[p]) > 0)
                min_index = p;
        }
        mpz_set(numbers[i], queue[min_index]);
    }
    free_numbers(queue, num_primes);
    free(primes);
    free(index);
    return numbers;
}

void print_nsmooth_numbers(uint32_t n, uint32_t begin, uint32_t count) {
    uint32_t num = begin + count;
    mpz_t* numbers = find_nsmooth_numbers(n, num);
    printf("%u: ", n);
    mpz_out_str(stdout, 10, numbers[begin]);
    for (uint32_t i = 1; i < count; ++i) {
        printf(", ");
        mpz_out_str(stdout, 10, numbers[begin + i]);
    }
    printf("\n");
    free_numbers(numbers, num);
}

int main() {
    printf("First 25 n-smooth numbers for n = 2 -> 29:\n");
    for (uint32_t n = 2; n <= 29; ++n) {
        if (is_prime(n))
            print_nsmooth_numbers(n, 0, 25);
    }
    printf("\n3 n-smooth numbers starting from 3000th for n = 3 -> 29:\n");
    for (uint32_t n = 3; n <= 29; ++n) {
        if (is_prime(n))
            print_nsmooth_numbers(n, 2999, 3);
    }
    printf("\n20 n-smooth numbers starting from 30,000th for n = 503 -> 521:\n");
    for (uint32_t n = 503; n <= 521; ++n) {
        if (is_prime(n))
            print_nsmooth_numbers(n, 29999, 20);
    }
    return 0;
}
Output:
First 25 n-smooth numbers for n = 2 -> 29:
2: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216
3: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192
5: 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
7: 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
11: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32
13: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28
17: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27
19: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26
23: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25
29: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25

3 n-smooth numbers starting from 3000th for n = 3 -> 29:
3: 91580367978306252441724649472, 92829823186414819915547541504, 94096325042746502515294076928
5: 278942752080, 279936000000, 281250000000
7: 50176000, 50331648, 50388480
11: 2112880, 2116800, 2117016
13: 390000, 390390, 390625
17: 145800, 145860, 146016
19: 74256, 74358, 74360
23: 46552, 46575, 46585
29: 33516, 33524, 33534

20 n-smooth numbers starting from 30,000th for n = 503 -> 521:
503: 62913, 62914, 62916, 62918, 62920, 62923, 62926, 62928, 62930, 62933, 62935, 62937, 62944, 62946, 62951, 62952, 62953, 62957, 62959, 62964
509: 62601, 62602, 62604, 62607, 62608, 62609, 62611, 62618, 62620, 62622, 62624, 62625, 62626, 62628, 62629, 62634, 62640, 62643, 62645, 62646
521: 62287, 62288, 62291, 62292, 62300, 62304, 62307, 62308, 62310, 62315, 62320, 62321, 62322, 62325, 62328, 62329, 62330, 62331, 62335, 62336

C++[edit]

Translation of: D

The output for the 30,000 3-smooth numbers is not correct due to insuffiant bits to represent that actual value

#include <algorithm>
#include <iostream>
#include <vector>

std::vector<uint64_t> primes;
std::vector<uint64_t> smallPrimes;

template <typename T>
std::ostream &operator <<(std::ostream &os, const std::vector<T> &v) {
    auto it = v.cbegin();
    auto end = v.cend();

    os << '[';

    if (it != end) {
        os << *it;
        it = std::next(it);
    }

    for (; it != end; it = std::next(it)) {
        os << ", " << *it;
    }

    return os << ']';
}

bool isPrime(uint64_t value) {
    if (value < 2) return false;

    if (value % 2 == 0) return value == 2;
    if (value % 3 == 0) return value == 3;

    if (value % 5 == 0) return value == 5;
    if (value % 7 == 0) return value == 7;

    if (value % 11 == 0) return value == 11;
    if (value % 13 == 0) return value == 13;

    if (value % 17 == 0) return value == 17;
    if (value % 19 == 0) return value == 19;

    if (value % 23 == 0) return value == 23;

    uint64_t t = 29;
    while (t * t < value) {
        if (value % t == 0) return false;
        value += 2;

        if (value % t == 0) return false;
        value += 4;
    }

    return true;
}

void init() {
    primes.push_back(2);
    smallPrimes.push_back(2);

    uint64_t i = 3;
    while (i <= 521) {
        if (isPrime(i)) {
            primes.push_back(i);
            if (i <= 29) {
                smallPrimes.push_back(i);
            }
        }
        i += 2;
    }
}

std::vector<uint64_t> nSmooth(uint64_t n, size_t size) {
    if (n < 2 || n>521) {
        throw std::runtime_error("n must be between 2 and 521");
    }
    if (size <= 1) {
        throw std::runtime_error("size must be at least 1");
    }

    uint64_t bn = n;
    if (primes.cend() == std::find(primes.cbegin(), primes.cend(), bn)) {
        throw std::runtime_error("n must be a prime number");
    }

    std::vector<uint64_t> ns(size, 0);
    ns[0] = 1;

    std::vector<uint64_t> next;
    for (auto prime : primes) {
        if (prime > bn) {
            break;
        }
        next.push_back(prime);
    }

    std::vector<size_t> indicies(next.size(), 0);
    for (size_t m = 1; m < size; m++) {
        ns[m] = *std::min_element(next.cbegin(), next.cend());
        for (size_t i = 0; i < indicies.size(); i++) {
            if (ns[m] == next[i]) {
                indicies[i]++;
                next[i] = primes[i] * ns[indicies[i]];
            }
        }
    }

    return ns;
}

int main() {
    init();

    for (auto i : smallPrimes) {
        std::cout << "The first " << i << "-smooth numbers are:\n";
        std::cout << nSmooth(i, 25) << '\n';
        std::cout << '\n';
    }

    // there is not enough bits to fully represent the 3-smooth numbers
    for (size_t i = 0; i < smallPrimes.size(); i++) {
        if (i < 1) continue;
        auto p = smallPrimes[i];

        auto v = nSmooth(p, 3002);
        v.erase(v.begin(), v.begin() + 2999);

        std::cout << "The 30,000th to 30,019th " << p << "-smooth numbers are:\n";
        std::cout << v << '\n';
        std::cout << '\n';
    }

    return 0;
}
Output:
The first 2-smooth numbers are:
[1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216]

The first 3-smooth numbers are:
[1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192]

The first 5-smooth numbers are:
[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]

The first 7-smooth numbers are:
[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]

The first 11-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32]

The first 13-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28]

The first 17-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27]

The first 19-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26]

The first 23-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]

The first 29-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]

The 30,000th to 30,019th 3-smooth numbers are:
[15283969189597937664, 12248739612010217472, 0]

The 30,000th to 30,019th 5-smooth numbers are:
[278942752080, 279936000000, 281250000000]

The 30,000th to 30,019th 7-smooth numbers are:
[50176000, 50331648, 50388480]

The 30,000th to 30,019th 11-smooth numbers are:
[2112880, 2116800, 2117016]

The 30,000th to 30,019th 13-smooth numbers are:
[390000, 390390, 390625]

The 30,000th to 30,019th 17-smooth numbers are:
[145800, 145860, 146016]

The 30,000th to 30,019th 19-smooth numbers are:
[74256, 74358, 74360]

The 30,000th to 30,019th 23-smooth numbers are:
[46552, 46575, 46585]

The 30,000th to 30,019th 29-smooth numbers are:
[33516, 33524, 33534]

C#[edit]

Translation of: D
using System;
using System.Collections.Generic;
using System.Linq;
using System.Numerics;

namespace NSmooth {
    class Program {
        static readonly List<BigInteger> primes = new List<BigInteger>();
        static readonly List<int> smallPrimes = new List<int>();

        static Program() {
            primes.Add(2);
            smallPrimes.Add(2);

            BigInteger i = 3;
            while (i <= 521) {
                if (IsPrime(i)) {
                    primes.Add(i);
                    if (i <= 29) {
                        smallPrimes.Add((int)i);
                    }
                }
                i += 2;
            }
        }

        static bool IsPrime(BigInteger value) {
            if (value < 2) return false;

            if (value % 2 == 0) return value == 2;
            if (value % 3 == 0) return value == 3;

            if (value % 5 == 0) return value == 5;
            if (value % 7 == 0) return value == 7;

            if (value % 11 == 0) return value == 11;
            if (value % 13 == 0) return value == 13;

            if (value % 17 == 0) return value == 17;
            if (value % 19 == 0) return value == 19;

            if (value % 23 == 0) return value == 23;

            BigInteger t = 29;
            while (t * t < value) {
                if (value % t == 0) return false;
                value += 2;

                if (value % t == 0) return false;
                value += 4;
            }

            return true;
        }

        static List<BigInteger> NSmooth(int n, int size) {
            if (n < 2 || n > 521) {
                throw new ArgumentOutOfRangeException("n");
            }
            if (size < 1) {
                throw new ArgumentOutOfRangeException("size");
            }

            BigInteger bn = n;
            bool ok = false;
            foreach (var prime in primes) {
                if (bn == prime) {
                    ok = true;
                    break;
                }
            }
            if (!ok) {
                throw new ArgumentException("must be a prime number", "n");
            }

            BigInteger[] ns = new BigInteger[size];
            ns[0] = 1;
            for (int i = 1; i < size; i++) {
                ns[i] = 0;
            }

            List<BigInteger> next = new List<BigInteger>();
            foreach (var prime in primes) {
                if (prime > bn) {
                    break;
                }
                next.Add(prime);
            }

            int[] indices = new int[next.Count];
            for (int i = 0; i < indices.Length; i++) {
                indices[i] = 0;
            }
            for (int m = 1; m < size; m++) {
                ns[m] = next.Min();
                for (int i = 0; i < indices.Length; i++) {
                    if (ns[m] == next[i]) {
                        indices[i]++;
                        next[i] = primes[i] * ns[indices[i]];
                    }
                }
            }

            return ns.ToList();
        }

        static void Println<T>(IEnumerable<T> nums) {
            Console.Write('[');

            var it = nums.GetEnumerator();
            if (it.MoveNext()) {
                Console.Write(it.Current);
            }
            while (it.MoveNext()) {
                Console.Write(", ");
                Console.Write(it.Current);
            }

            Console.WriteLine(']');
        }

        static void Main() {
            foreach (var i in smallPrimes) {
                Console.WriteLine("The first {0}-smooth numbers are:", i);
                Println(NSmooth(i, 25));
                Console.WriteLine();
            }
            foreach (var i in smallPrimes.Skip(1)) {
                Console.WriteLine("The 3,000 to 3,202 {0}-smooth numbers are:", i);
                Println(NSmooth(i, 3_002).Skip(2_999));
                Console.WriteLine();
            }
            foreach (var i in new int[] { 503, 509, 521 }) {
                Console.WriteLine("The 30,000 to 3,019 {0}-smooth numbers are:", i);
                Println(NSmooth(i, 30_019).Skip(29_999));
                Console.WriteLine();
            }
        }
    }
}
Output:
The first 2-smooth numbers are:
[1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216]

The first 3-smooth numbers are:
[1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192]

The first 5-smooth numbers are:
[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]

The first 7-smooth numbers are:
[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]

The first 11-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32]

The first 13-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28]

The first 17-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27]

The first 19-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26]

The first 23-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]

The first 29-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]

The 3,000 to 3,202 3-smooth numbers are:
[91580367978306252441724649472, 92829823186414819915547541504, 94096325042746502515294076928]

The 3,000 to 3,202 5-smooth numbers are:
[278942752080, 279936000000, 281250000000]

The 3,000 to 3,202 7-smooth numbers are:
[50176000, 50331648, 50388480]

The 3,000 to 3,202 11-smooth numbers are:
[2112880, 2116800, 2117016]

The 3,000 to 3,202 13-smooth numbers are:
[390000, 390390, 390625]

The 3,000 to 3,202 17-smooth numbers are:
[145800, 145860, 146016]

The 3,000 to 3,202 19-smooth numbers are:
[74256, 74358, 74360]

The 3,000 to 3,202 23-smooth numbers are:
[46552, 46575, 46585]

The 3,000 to 3,202 29-smooth numbers are:
[33516, 33524, 33534]

The 30,000 to 3,019 503-smooth numbers are:
[62913, 62914, 62916, 62918, 62920, 62923, 62926, 62928, 62930, 62933, 62935, 62937, 62944, 62946, 62951, 62952, 62953, 62957, 62959, 62964]

The 30,000 to 3,019 509-smooth numbers are:
[62601, 62602, 62604, 62607, 62608, 62609, 62611, 62618, 62620, 62622, 62624, 62625, 62626, 62628, 62629, 62634, 62640, 62643, 62645, 62646]

The 30,000 to 3,019 521-smooth numbers are:
[62287, 62288, 62291, 62292, 62300, 62304, 62307, 62308, 62310, 62315, 62320, 62321, 62322, 62325, 62328, 62329, 62330, 62331, 62335, 62336]

Crystal[edit]

Translation of: Ruby
require "big"

def prime?(n) # P3 Prime Generator primality test
  return false unless (n | 1 == 3 if n < 5) || (n % 6) | 4 == 5
  sqrt_n = Math.isqrt(n)  # For Crystal < 1.2.0 use Math.sqrt(n).to_i
  pc = typeof(n).new(5)
  while pc <= sqrt_n
    return false if n % pc == 0 || n % (pc + 2) == 0
    pc += 6
  end
  true
end

def gen_primes(a, b)
    (a..b).select { |pc| pc if prime? pc }
end

def nsmooth(n, limit)
    raise "Exception(n or limit)" if n < 2 || n > 521 || limit < 1
    raise "Exception(must be a prime number: n)" unless prime? n
    
    primes = gen_primes(2, n)
    ns = [0.to_big_i] * limit
    ns[0] = 1.to_big_i
    nextp = primes[0..primes.index(n)].map { |prm| prm.to_big_i }

    indices = [0] * nextp.size
    (1...limit).each do |m|
        ns[m] = nextp.min
        (0...indices.size).each do |i|
            if ns[m] == nextp[i]
                indices[i] += 1
                nextp[i] = primes[i] * ns[indices[i]]
            end
        end
    end
    ns
end

gen_primes(2, 29).each do |prime|
    print "The first 25 #{prime}-smooth numbers are: \n"
    print nsmooth(prime, 25)
    puts
end
puts
gen_primes(3, 29).each do |prime|
    print "The 3000 to 3202 #{prime}-smooth numbers are: "
    print nsmooth(prime, 3002)[2999..]
    puts
end
puts
gen_primes(503, 521).each do |prime|
    print "The 30,000 to 30,019 #{prime}-smooth numbers are: \n"
    print nsmooth(prime, 30019)[29999..]
    puts
end
Output:
The first 25 2-smooth numbers are: 
[1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216]
The first 25 3-smooth numbers are: 
[1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192]
The first 25 5-smooth numbers are: 
[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]
The first 25 7-smooth numbers are: 
[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]
The first 25 11-smooth numbers are: 
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32]
The first 25 13-smooth numbers are: 
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28]
The first 25 17-smooth numbers are: 
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27]
The first 25 19-smooth numbers are: 
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26]
The first 25 23-smooth numbers are: 
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]
The first 25 29-smooth numbers are: 
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]

The 3000 to 3002 3-smooth numbers are: [91580367978306252441724649472, 92829823186414819915547541504, 94096325042746502515294076928]
The 3000 to 3002 5-smooth numbers are: [278942752080, 279936000000, 281250000000]
The 3000 to 3002 7-smooth numbers are: [50176000, 50331648, 50388480]
The 3000 to 3002 11-smooth numbers are: [2112880, 2116800, 2117016]
The 3000 to 3002 13-smooth numbers are: [390000, 390390, 390625]
The 3000 to 3002 17-smooth numbers are: [145800, 145860, 146016]
The 3000 to 3002 19-smooth numbers are: [74256, 74358, 74360]
The 3000 to 3002 23-smooth numbers are: [46552, 46575, 46585]
The 3000 to 3002 29-smooth numbers are: [33516, 33524, 33534]

The 30,000 to 30,019 503-smooth numbers are: 
[62913, 62914, 62916, 62918, 62920, 62923, 62926, 62928, 62930, 62933, 62935, 62937, 62944, 62946, 62951, 62952, 62953, 62957, 62959, 62964]
The 30,000 to 30,019 509-smooth numbers are: 
[62601, 62602, 62604, 62607, 62608, 62609, 62611, 62618, 62620, 62622, 62624, 62625, 62626, 62628, 62629, 62634, 62640, 62643, 62645, 62646]
The 30,000 to 30,019 521-smooth numbers are: 
[62287, 62288, 62291, 62292, 62300, 62304, 62307, 62308, 62310, 62315, 62320, 62321, 62322, 62325, 62328, 62329, 62330, 62331, 62335, 62336]

D[edit]

Translation of: Kotlin
import std.algorithm;
import std.bigint;
import std.exception;
import std.range;
import std.stdio;

BigInt[] primes;
int[] smallPrimes;

bool isPrime(BigInt value) {
    if (value < 2) return false;

    if (value % 2 == 0) return value == 2;
    if (value % 3 == 0) return value == 3;

    if (value % 5 == 0) return value == 5;
    if (value % 7 == 0) return value == 7;

    if (value % 11 == 0) return value == 11;
    if (value % 13 == 0) return value == 13;

    if (value % 17 == 0) return value == 17;
    if (value % 19 == 0) return value == 19;

    if (value % 23 == 0) return value == 23;

    BigInt t = 29;
    while (t * t < value) {
        if (value % t == 0) return false;
        value += 2;

        if (value % t == 0) return false;
        value += 4;
    }

    return true;
}

// cache all primes up to 521
void init() {
    primes ~= BigInt(2);
    smallPrimes ~= 2;

    BigInt i = 3;
    while (i <= 521) {
        if (isPrime(i)) {
            primes ~= i;
            if (i <= 29) {
                smallPrimes ~= i.toInt;
            }
        }
        i += 2;
    }
}

BigInt[] nSmooth(int n, int size)
in {
    enforce(n >= 2 && n <= 521, "n must be between 2 and 521");
    enforce(size > 1, "size must be at least 1");
}
do {
    BigInt bn = n;
    bool ok = false;
    foreach (prime; primes) {
        if (bn == prime) {
            ok = true;
            break;
        }
    }
    enforce(ok, "n must be a prime number");

    BigInt[] ns;
    ns.length = size;
    ns[] = BigInt(0);
    ns[0] = 1;

    BigInt[] next;
    foreach(prime; primes) {
        if (prime > bn) {
            break;
        }
        next ~= prime;
    }

    int[] indicies;
    indicies.length = next.length;
    indicies[] = 0;
    foreach (m; 1 .. size) {
        ns[m] = next.reduce!min;
        foreach (i,v; indicies) {
            if (ns[m] == next[i]) {
                indicies[i]++;
                next[i] = primes[i] * ns[indicies[i]];
            }
        }
    }

    return ns;
}

void main() {
    init();

    foreach (i; smallPrimes) {
        writeln("The first ", i, "-smooth numbers are:");
        writeln(nSmooth(i, 25));
        writeln;
    }
    foreach (i; smallPrimes.drop(1)) {
        writeln("The 3,000th to 3,202 ", i, "-smooth numbers are:");
        writeln(nSmooth(i, 3_002).drop(2_999));
        writeln;
    }
    foreach (i; [503, 509, 521]) {
        writeln("The 30,000th to 30,019 ", i, "-smooth numbers are:");
        writeln(nSmooth(i, 30_019).drop(29_999));
        writeln;
    }
}
Output:
The first 2-smooth numbers are:
[1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216]

The first 3-smooth numbers are:
[1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192]

The first 5-smooth numbers are:
[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]

The first 7-smooth numbers are:
[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]

The first 11-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32]

The first 13-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28]

The first 17-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27]

The first 19-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26]

The first 23-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]

The first 29-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]

The 3,000th to 3,202 3-smooth numbers are:
[91580367978306252441724649472, 92829823186414819915547541504, 94096325042746502515294076928]

The 3,000th to 3,202 5-smooth numbers are:
[278942752080, 279936000000, 281250000000]

The 3,000th to 3,202 7-smooth numbers are:
[50176000, 50331648, 50388480]

The 3,000th to 3,202 11-smooth numbers are:
[2112880, 2116800, 2117016]

The 3,000th to 3,202 13-smooth numbers are:
[390000, 390390, 390625]

The 3,000th to 3,202 17-smooth numbers are:
[145800, 145860, 146016]

The 3,000th to 3,202 19-smooth numbers are:
[74256, 74358, 74360]

The 3,000th to 3,202 23-smooth numbers are:
[46552, 46575, 46585]

The 3,000th to 3,202 29-smooth numbers are:
[33516, 33524, 33534]

The 30,000th to 30,019 503-smooth numbers are:
[62913, 62914, 62916, 62918, 62920, 62923, 62926, 62928, 62930, 62933, 62935, 62937, 62944, 62946, 62951, 62952, 62953, 62957, 62959, 62964]

The 30,000th to 30,019 509-smooth numbers are:
[62601, 62602, 62604, 62607, 62608, 62609, 62611, 62618, 62620, 62622, 62624, 62625, 62626, 62628, 62629, 62634, 62640, 62643, 62645, 62646]

The 30,000th to 30,019 521-smooth numbers are:
[62287, 62288, 62291, 62292, 62300, 62304, 62307, 62308, 62310, 62315, 62320, 62321, 62322, 62325, 62328, 62329, 62330, 62331, 62335, 62336]

Delphi[edit]

Thanks for Rudy Velthuis[1]
Translation of: D
program N_smooth_numbers;

{$APPTYPE CONSOLE}
{$R *.res}

uses
  System.SysUtils,
  System.Generics.Collections,
  Velthuis.BigIntegers;

var
  primes: TList<BigInteger>;
  smallPrimes: TList<Integer>;

function IsPrime(value: BigInteger): Boolean;
var
  v: BigInteger;
begin
  if value < 2 then
    exit(False);

  for v in [2, 3, 5, 7, 11, 13, 17, 19, 23] do
  begin
    if (value mod v) = 0 then
      exit(value = v);
  end;

  v := 29;

  while v * v < value do
  begin
    if (value mod v) = 0 then
      exit(False);
    inc(value, 2);
    if (value mod v) = 0 then
      exit(False);
    inc(v, 4);
  end;

  Result := True;
end;

function Min(values: TList<BigInteger>): BigInteger;
var
  value: BigInteger;
begin
  if values.Count = 0 then
    exit(0);

  Result := values[0];
  for value in values do
  begin
    if value < Result then
      result := value;
  end;
end;

function NSmooth(n, size: Integer): TList<BigInteger>;
var
  bn, p: BigInteger;
  ok: Boolean;
  i: Integer;
  next: TList<BigInteger>;
  indices: TList<Integer>;
  m: Integer;
begin
  Result := TList<BigInteger>.Create;
  if (n < 2) or (n > 521) then
    raise Exception.Create('Argument out of range: "n"');

  if (size < 1) then
    raise Exception.Create('Argument out of range: "size"');

  bn := n;
  ok := false;
  for p in primes do
  begin
    ok := bn = p;
    if ok then
      break;
  end;

  if not ok then
    raise Exception.Create('"n" must be a prime number');

  Result.Add(1);

  for i := 1 to size - 1 do
    Result.Add(0);

  next := TList<BigInteger>.Create;

  for p in primes do
  begin
    if p > bn then
      Break;
    next.Add(p);
  end;

  indices := TList<Integer>.Create;
  for i := 0 to next.Count - 1 do
    indices.Add(0);

  for m := 1 to size - 1 do
  begin
    Result[m] := Min(next);
    for i := 0 to indices.Count - 1 do
      if Result[m] = next[i] then
      begin
        indices[i] := indices[i] + 1;
        next[i] := primes[i] * Result[indices[i]];
      end;
  end;

  indices.Free;
  next.Free;
end;

procedure Init();
var
  i: BigInteger;
begin
  primes := TList<BigInteger>.Create;
  smallPrimes := TList<Integer>.Create;
  primes.Add(2);
  smallPrimes.Add(2);

  i := 3;
  while i <= 521 do
  begin
    if IsPrime(i) then
    begin

      primes.Add(i);
      if i <= 29 then
        smallPrimes.Add(Integer(i));
    end;
    inc(i, 2);
  end;

end;

procedure Println(values: TList<BigInteger>; CanFree: Boolean = False);
var
  value: BigInteger;
begin
  Write('[');
  for value in values do
    Write(value.ToString, ', ');
  Writeln(']'#10);

  if CanFree then
    values.Free;
end;

procedure Finish();
begin
  primes.Free;
  smallPrimes.Free;
end;

var
  p: Integer;
  ns: TList<BigInteger>;

const
  RANGE_3: array[0..2] of integer = (503, 509, 521);

begin
  Init;
  for p in smallPrimes do
  begin
    Writeln('The first ', p, '-smooth numbers are:');
    Println(NSmooth(p, 25), True);
  end;

  smallPrimes.Delete(0);
  for p in smallPrimes do
  begin
    Writeln('The 3,000 to 3,202 ', p, '-smooth numbers are:');
    ns := nSmooth(p, 3002);
    ns.DeleteRange(0, 2999);
    println(ns, True);
  end;

  for p in RANGE_3 do
  begin
    Writeln('The 3,000 to 3,019 ', p, '-smooth numbers are:');
    ns := nSmooth(p, 30019);
    ns.DeleteRange(0, 29999);
    println(ns, True);
  end;

  Finish;
  Readln;
end.

Factor[edit]

USING: deques dlists formatting fry io kernel locals make math
math.order math.primes math.text.english namespaces prettyprint
sequences tools.memory.private ;
IN: rosetta-code.n-smooth-numbers

SYMBOL: primes
 
: ns ( n -- seq )
    primes-upto [ primes set ] [ length [ 1 1dlist ] replicate ]
    bi ;
 
: enqueue ( n seq -- )
    [ primes get ] 2dip [ '[ _ * ] map ] dip [ push-back ] 2each
    ;
 
: next ( seq -- n )
    dup [ peek-front ] map infimum
    [ '[ dup peek-front _ = [ pop-front* ] [ drop ] if ] each ]
    [ swap enqueue ] [ nip ] 2tri ;
 
: next-n ( seq n -- seq )
    swap '[ _ [ _ next , ] times ] { } make ;

:: n-smooth ( n from to -- seq )
    n ns to next-n to from - 1 + tail* ;
 
:: show-smooth ( plo phi lo hi -- )
    plo phi primes-between [
        :> p lo commas lo ordinal-suffix hi commas hi
        ordinal-suffix p "%s%s through %s%s %d-smooth numbers: "
        printf p lo hi n-smooth [ pprint bl ] each nl        
    ] each ;

: smooth-numbers-demo ( -- )
    2 29 1 25 show-smooth nl
    3 29 3000 3002 show-smooth nl
    503 521 30,000 30,019 show-smooth ;

MAIN: smooth-numbers-demo
Output:
1st through 25th 2-smooth numbers: 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576 2097152 4194304 8388608 16777216 
1st through 25th 3-smooth numbers: 1 2 3 4 6 8 9 12 16 18 24 27 32 36 48 54 64 72 81 96 108 128 144 162 192 
1st through 25th 5-smooth numbers: 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 
1st through 25th 7-smooth numbers: 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 
1st through 25th 11-smooth numbers: 1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 18 20 21 22 24 25 27 28 30 32 
1st through 25th 13-smooth numbers: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 21 22 24 25 26 27 28 
1st through 25th 17-smooth numbers: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20 21 22 24 25 26 27 
1st through 25th 19-smooth numbers: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 24 25 26 
1st through 25th 23-smooth numbers: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 
1st through 25th 29-smooth numbers: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 

3,000th through 3,002nd 3-smooth numbers: 91580367978306252441724649472 92829823186414819915547541504 94096325042746502515294076928 
3,000th through 3,002nd 5-smooth numbers: 278942752080 279936000000 281250000000 
3,000th through 3,002nd 7-smooth numbers: 50176000 50331648 50388480 
3,000th through 3,002nd 11-smooth numbers: 2112880 2116800 2117016 
3,000th through 3,002nd 13-smooth numbers: 390000 390390 390625 
3,000th through 3,002nd 17-smooth numbers: 145800 145860 146016 
3,000th through 3,002nd 19-smooth numbers: 74256 74358 74360 
3,000th through 3,002nd 23-smooth numbers: 46552 46575 46585 
3,000th through 3,002nd 29-smooth numbers: 33516 33524 33534 

30,000th through 30,019th 503-smooth numbers: 62913 62914 62916 62918 62920 62923 62926 62928 62930 62933 62935 62937 62944 62946 62951 62952 62953 62957 62959 62964 
30,000th through 30,019th 509-smooth numbers: 62601 62602 62604 62607 62608 62609 62611 62618 62620 62622 62624 62625 62626 62628 62629 62634 62640 62643 62645 62646 
30,000th through 30,019th 521-smooth numbers: 62287 62288 62291 62292 62300 62304 62307 62308 62310 62315 62320 62321 62322 62325 62328 62329 62330 62331 62335 62336 

Go[edit]

package main

import (
    "fmt"
    "log"
    "math/big"
)

var (
    primes      []*big.Int
    smallPrimes []int
)

// cache all primes up to 521
func init() {
    two := big.NewInt(2)
    three := big.NewInt(3)
    p521 := big.NewInt(521)
    p29 := big.NewInt(29)
    primes = append(primes, two)
    smallPrimes = append(smallPrimes, 2)
    for i := three; i.Cmp(p521) <= 0; i.Add(i, two) {
        if i.ProbablyPrime(0) {
            primes = append(primes, new(big.Int).Set(i))
            if i.Cmp(p29) <= 0 {
                smallPrimes = append(smallPrimes, int(i.Int64()))
            }
        }
    }
}

func min(bs []*big.Int) *big.Int {
    if len(bs) == 0 {
        log.Fatal("slice must have at least one element")
    }
    res := bs[0]
    for _, i := range bs[1:] {
        if i.Cmp(res) < 0 {
            res = i
        }
    }
    return res
}

func nSmooth(n, size int) []*big.Int {
    if n < 2 || n > 521 {
        log.Fatal("n must be between 2 and 521")
    }
    if size < 1 {
        log.Fatal("size must be at least 1")
    }
    bn := big.NewInt(int64(n))
    ok := false
    for _, prime := range primes {
        if bn.Cmp(prime) == 0 {
            ok = true
            break
        }
    }
    if !ok {
        log.Fatal("n must be a prime number")
    }

    ns := make([]*big.Int, size)
    ns[0] = big.NewInt(1)
    var next []*big.Int
    for i := 0; i < len(primes); i++ {
        if primes[i].Cmp(bn) > 0 {
            break
        }
        next = append(next, new(big.Int).Set(primes[i]))
    }
    indices := make([]int, len(next))
    for m := 1; m < size; m++ {
        ns[m] = new(big.Int).Set(min(next))
        for i := 0; i < len(indices); i++ {
            if ns[m].Cmp(next[i]) == 0 {
                indices[i]++
                next[i].Mul(primes[i], ns[indices[i]])
            }
        }
    }
    return ns
}

func main() {
    for _, i := range smallPrimes {
        fmt.Printf("The first 25 %d-smooth numbers are:\n", i)
        fmt.Println(nSmooth(i, 25), "\n")
    }
    for _, i := range smallPrimes[1:] {
        fmt.Printf("The 3,000th to 3,202nd %d-smooth numbers are:\n", i)
        fmt.Println(nSmooth(i, 3002)[2999:], "\n")
    }
    for _, i := range []int{503, 509, 521} {
        fmt.Printf("The 30,000th to 30,019th %d-smooth numbers are:\n", i)
        fmt.Println(nSmooth(i, 30019)[29999:], "\n")
    }
}
Output:
The first 25 2-smooth numbers are:
[1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576 2097152 4194304 8388608 16777216] 

The first 25 3-smooth numbers are:
[1 2 3 4 6 8 9 12 16 18 24 27 32 36 48 54 64 72 81 96 108 128 144 162 192] 

The first 25 5-smooth numbers are:
[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] 

The first 25 7-smooth numbers are:
[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] 

The first 25 11-smooth numbers are:
[1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 18 20 21 22 24 25 27 28 30 32] 

The first 25 13-smooth numbers are:
[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 21 22 24 25 26 27 28] 

The first 25 17-smooth numbers are:
[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20 21 22 24 25 26 27] 

The first 25 19-smooth numbers are:
[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 24 25 26] 

The first 25 23-smooth numbers are:
[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25] 

The first 25 29-smooth numbers are:
[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25] 

The 3,000th to 3,202nd 3-smooth numbers are:
[91580367978306252441724649472 92829823186414819915547541504 94096325042746502515294076928] 

The 3,000th to 3,202nd 5-smooth numbers are:
[278942752080 279936000000 281250000000] 

The 3,000th to 3,202nd 7-smooth numbers are:
[50176000 50331648 50388480] 

The 3,000th to 3,202nd 11-smooth numbers are:
[2112880 2116800 2117016] 

The 3,000th to 3,202nd 13-smooth numbers are:
[390000 390390 390625] 

The 3,000th to 3,202nd 17-smooth numbers are:
[145800 145860 146016] 

The 3,000th to 3,202nd 19-smooth numbers are:
[74256 74358 74360] 

The 3,000th to 3,202nd 23-smooth numbers are:
[46552 46575 46585] 

The 3,000th to 3,202nd 29-smooth numbers are:
[33516 33524 33534] 

The 30,000th to 30,019th 503-smooth numbers are:
[62913 62914 62916 62918 62920 62923 62926 62928 62930 62933 62935 62937 62944 62946 62951 62952 62953 62957 62959 62964] 

The 30,000th to 30,019th 509-smooth numbers are:
[62601 62602 62604 62607 62608 62609 62611 62618 62620 62622 62624 62625 62626 62628 62629 62634 62640 62643 62645 62646] 

The 30,000th to 30,019th 521-smooth numbers are:
[62287 62288 62291 62292 62300 62304 62307 62308 62310 62315 62320 62321 62322 62325 62328 62329 62330 62331 62335 62336] 

Haskell[edit]

The solution is based on the hamming numbers solution Hamming_numbers#Avoiding_generation_of_duplicates.
Uses Library Data.Number.Primes: http://hackage.haskell.org/package/primes-0.2.1.0/docs/Data-Numbers-Primes.html

import Data.Numbers.Primes (primes)
import Text.Printf (printf)

 
merge :: Ord a => [a] -> [a] -> [a]
merge [] b = b
merge a@(x:xs) b@(y:ys) | x < y     = x : merge xs b
                        | otherwise = y : merge a ys
 
nSmooth :: Integer -> [Integer]
nSmooth p = 1 : foldr u [] factors
 where
   factors = takeWhile (<=p) primes
   u n s = r
    where r = merge s (map (n*) (1:r))

main :: IO ()
main = do
  mapM_ (printf "First 25 %d-smooth:\n%s\n\n" <*> showTwentyFive) firstTenPrimes
  mapM_
    (printf "The 3,000 to 3,202 %d-smooth numbers are:\n%s\n\n" <*> showRange1)
    firstTenPrimes
  mapM_
    (printf "The 30,000 to 30,019 %d-smooth numbers are:\n%s\n\n" <*> showRange2)
    [503, 509, 521]
  where
    firstTenPrimes = take 10 primes
    showTwentyFive = show . take 25 . nSmooth
    showRange1 = show . ((<$> [2999 .. 3001]) . (!!) . nSmooth)
    showRange2 = show . ((<$> [29999 .. 30018]) . (!!) . nSmooth)
Output:
First 25 2-smooth:
[1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,65536,131072,262144,524288,1048576,2097152,4194304,8388608,16777216]

First 25 3-smooth:
[1,2,3,4,6,8,9,12,16,18,24,27,32,36,48,54,64,72,81,96,108,128,144,162,192]

First 25 5-smooth:
[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]

First 25 7-smooth:
[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]

First 25 11-smooth:
[1,2,3,4,5,6,7,8,9,10,11,12,14,15,16,18,20,21,22,24,25,27,28,30,32]

First 25 13-smooth:
[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,18,20,21,22,24,25,26,27,28]

First 25 17-smooth:
[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,20,21,22,24,25,26,27]

First 25 19-smooth:
[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,24,25,26]

First 25 23-smooth:
[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]

First 25 29-smooth:
[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]

The 3,000 to 3,202 2-smooth numbers are:
[615115961080558588465779406638376257320356947868416857883059014580029400307336474387680033919296729791214824627025902454256442090449118411792541241032674165617479675177922508706511660055680333461312364119878440208217239157846837506706545378604345188396648329405331470912246744225863252651856458002673373954311851336740459676968406552868310201176372388451920238941825550161204650991744181901465270241243954881742049126970364342566022204431867377135606296235889321974743344255860525780985216390373727411888404232090348551541930906092174282761370097898341311102755922756040276005155025127900794674822964000566872737110357506841706953771389531879916938050677117592122548335021080360314705790751185624004215223592421049305160290208996103331123664361061044256821841953835180104581326835320565468498501085250337750687361999383002913789650361626737445306125067585944587449539955645756199886936089259509114994688,1230231922161117176931558813276752514640713895736833715766118029160058800614672948775360067838593459582429649254051804908512884180898236823585082482065348331234959350355845017413023320111360666922624728239756880416434478315693675013413090757208690376793296658810662941824493488451726505303712916005346747908623702673480919353936813105736620402352744776903840477883651100322409301983488363802930540482487909763484098253940728685132044408863734754271212592471778643949486688511721051561970432780747454823776808464180697103083861812184348565522740195796682622205511845512080552010310050255801589349645928001133745474220715013683413907542779063759833876101354235184245096670042160720629411581502371248008430447184842098610320580417992206662247328722122088513643683907670360209162653670641130936997002170500675501374723998766005827579300723253474890612250135171889174899079911291512399773872178519018229989376,2460463844322234353863117626553505029281427791473667431532236058320117601229345897550720135677186919164859298508103609817025768361796473647170164964130696662469918700711690034826046640222721333845249456479513760832868956631387350026826181514417380753586593317621325883648986976903453010607425832010693495817247405346961838707873626211473240804705489553807680955767302200644818603966976727605861080964975819526968196507881457370264088817727469508542425184943557287898973377023442103123940865561494909647553616928361394206167723624368697131045480391593365244411023691024161104020620100511603178699291856002267490948441430027366827815085558127519667752202708470368490193340084321441258823163004742496016860894369684197220641160835984413324494657444244177027287367815340720418325307341282261873994004341001351002749447997532011655158601446506949781224500270343778349798159822583024799547744357038036459978752]

The 3,000 to 3,202 3-smooth numbers are:
[91580367978306252441724649472,92829823186414819915547541504,94096325042746502515294076928]

The 3,000 to 3,202 5-smooth numbers are:
[278942752080,279936000000,281250000000]

The 3,000 to 3,202 7-smooth numbers are:
[50176000,50331648,50388480]

The 3,000 to 3,202 11-smooth numbers are:
[2112880,2116800,2117016]

The 3,000 to 3,202 13-smooth numbers are:
[390000,390390,390625]

The 3,000 to 3,202 17-smooth numbers are:
[145800,145860,146016]

The 3,000 to 3,202 19-smooth numbers are:
[74256,74358,74360]

The 3,000 to 3,202 23-smooth numbers are:
[46552,46575,46585]

The 3,000 to 3,202 29-smooth numbers are:
[33516,33524,33534]

The 30,000 to 30,019 503-smooth numbers are:
[62913,62914,62916,62918,62920,62923,62926,62928,62930,62933,62935,62937,62944,62946,62951,62952,62953,62957,62959,62964]

The 30,000 to 30,019 509-smooth numbers are:
[62601,62602,62604,62607,62608,62609,62611,62618,62620,62622,62624,62625,62626,62628,62629,62634,62640,62643,62645,62646]

The 30,000 to 30,019 521-smooth numbers are:
[62287,62288,62291,62292,62300,62304,62307,62308,62310,62315,62320,62321,62322,62325,62328,62329,62330,62331,62335,62336]

./n-smooth  0.03s user 0.01s system 10% cpu 0.383 total

J[edit]

This solution involves sorting, duplicate removal, and limits on the number of preserved terms.

nsmooth=: dyad define  NB. TALLY nsmooth N
 factors=. x: i.@:>:&.:(p:inv) y
 smoothies=. , 1x
 result=. , i. 0x
 while. x > # result do.
  mn =. {. smoothies
  smoothies =. ({.~ (x <. #)) ~. /:~ (}. smoothies) , mn * factors
  result=. result , mn  
 end.
)
   25 (] ; nsmooth)&> p: i. 10
┌──┬────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┐
│2 │1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576 2097152 4194304 8388608 16777216│
├──┼────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┤
│3 │1 2 3 4 6 8 9 12 16 18 24 27 32 36 48 54 64 72 81 96 108 128 144 162 192                                                        │
├──┼────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┤
│5 │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                                                              │
├──┼────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┤
│7 │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                                                               │
├──┼────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┤
│11│1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 18 20 21 22 24 25 27 28 30 32                                                               │
├──┼────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┤
│13│1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 21 22 24 25 26 27 28                                                               │
├──┼────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┤
│17│1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20 21 22 24 25 26 27                                                               │
├──┼────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┤
│19│1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 24 25 26                                                               │
├──┼────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┤
│23│1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25                                                               │
├──┼────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┤
│29│1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25                                                               │
└──┴────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┘

   3002 (,.@:] ; (_3 {. nsmooth)&>) p: }. i. 10
┌──┬─────────────────────────────────────────────────────────────────────────────────────────┐
│ 3│91580367978306252441724649472 92829823186414819915547541504 94096325042746502515294076928│
│ 5│                 278942752080                  279936000000                  281250000000│
│ 7│                     50176000                      50331648                      50388480│
│11│                      2112880                       2116800                       2117016│
│13│                       390000                        390390                        390625│
│17│                       145800                        145860                        146016│
│19│                        74256                         74358                         74360│
│23│                        46552                         46575                         46585│
│29│                        33516                         33524                         33534│
└──┴─────────────────────────────────────────────────────────────────────────────────────────┘

   (i.3)&+&.:(p:inv)503
503 509 521

   (30000+20-1) (,.@:] ; (_20 {. nsmooth)&>) (i.3)&+&.:(p:inv)503
┌───┬───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┐
│503│62913 62914 62916 62918 62920 62923 62926 62928 62930 62933 62935 62937 62944 62946 62951 62952 62953 62957 62959 62964│
│509│62601 62602 62604 62607 62608 62609 62611 62618 62620 62622 62624 62625 62626 62628 62629 62634 62640 62643 62645 62646│
│521│62287 62288 62291 62292 62300 62304 62307 62308 62310 62315 62320 62321 62322 62325 62328 62329 62330 62331 62335 62336│
└───┴───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┘

Java[edit]

import java.math.BigInteger;
import java.util.ArrayList;
import java.util.List;

public class NSmoothNumbers {

    public static void main(String[] args) {
        System.out.printf("show the first 25 n-smooth numbers for n = 2 through n = 29%n");
        int max = 25;
        List<BigInteger> primes = new ArrayList<>();
        for ( int n = 2 ; n <= 29 ; n++ ) {
            if ( isPrime(n) ) {
                primes.add(BigInteger.valueOf(n));
                System.out.printf("The first %d %d-smooth numbers:%n", max, n);
                BigInteger[] humble = nSmooth(max, primes.toArray(new BigInteger[0]));
                for ( int i = 0 ; i < max ; i++ ) {
                    System.out.printf("%s ", humble[i]);
                }
                System.out.printf("%n%n");
            }
        }
        
        System.out.printf("show three numbers starting with 3,000 for n-smooth numbers for n = 3 through n = 29%n");
        int count = 3;
        max = 3000 + count - 1;
        primes = new ArrayList<>();
        primes.add(BigInteger.valueOf(2));
        for ( int n = 3 ; n <= 29 ; n++ ) {
            if ( isPrime(n) ) {
                primes.add(BigInteger.valueOf(n));
                System.out.printf("The %d through %d %d-smooth numbers:%n", max-count+1, max, n);
                BigInteger[] nSmooth = nSmooth(max, primes.toArray(new BigInteger[0]));
                for ( int i = max-count ; i < max ; i++ ) {
                    System.out.printf("%s ", nSmooth[i]);
                }
                System.out.printf("%n%n");
            }
        }
        
        System.out.printf("Show twenty numbers starting with 30,000 n-smooth numbers for n=503 through n=521%n");
        count = 20;
        max = 30000 + count - 1;
        primes = new ArrayList<>();
        for ( int n = 2 ; n <= 521 ; n++ ) {
            if ( isPrime(n) ) {
                primes.add(BigInteger.valueOf(n));
                if ( n >= 503 && n <= 521 ) {
                    System.out.printf("The %d through %d %d-smooth numbers:%n", max-count+1, max, n);
                    BigInteger[] nSmooth = nSmooth(max, primes.toArray(new BigInteger[0]));
                    for ( int i = max-count ; i < max ; i++ ) {
                        System.out.printf("%s ", nSmooth[i]);
                    }
                    System.out.printf("%n%n");
                }
            }
        }

    }

    private static final boolean isPrime(long test) {
        if ( test == 2 ) {
            return true;
        }
        if ( test % 2 == 0 ) return false;
        for ( long i = 3 ; i <= Math.sqrt(test) ; i += 2 ) {
            if ( test % i == 0 ) {
                return false;
            }
        }
        return true;
    }

    private static BigInteger[] nSmooth(int n, BigInteger[] primes) {
        int size = primes.length;
        BigInteger[] test = new BigInteger[size];
        for ( int i = 0 ; i < size ; i++ ) {
            test[i] = primes[i];
        }
        BigInteger[] results = new BigInteger[n];
        results[0] = BigInteger.ONE;
        
        int[] indexes = new int[size];
        for ( int i = 0 ; i < size ; i++ ) {
            indexes[i] = 0;
        }
        
        for ( int index = 1 ; index < n ; index++ ) {
            BigInteger min = test[0];
            for ( int i = 1 ; i < size ; i++ ) {
                min = min.min(test[i]);
            }
            results[index] = min;
            
            for ( int i = 0 ; i < size ; i++ ) {
                if ( results[index].compareTo(test[i]) == 0 ) {
                    indexes[i] = indexes[i] + 1;
                    test[i] = primes[i].multiply(results[indexes[i]]);
                }
            }
        }
        return results;
    }

}
Output:
show the first 25 n-smooth numbers for n = 2 through n = 29
The first 25 2-smooth numbers:
1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576 2097152 4194304 8388608 16777216 

The first 25 3-smooth numbers:
1 2 3 4 6 8 9 12 16 18 24 27 32 36 48 54 64 72 81 96 108 128 144 162 192 

The first 25 5-smooth numbers:
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 

The first 25 7-smooth numbers:
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 

The first 25 11-smooth numbers:
1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 18 20 21 22 24 25 27 28 30 32 

The first 25 13-smooth numbers:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 21 22 24 25 26 27 28 

The first 25 17-smooth numbers:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20 21 22 24 25 26 27 

The first 25 19-smooth numbers:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 24 25 26 

The first 25 23-smooth numbers:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 

The first 25 29-smooth numbers:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 

show three numbers starting with 3,000 for n-smooth numbers for n = 3 through n = 29
The 3000 through 3002 3-smooth numbers:
91580367978306252441724649472 92829823186414819915547541504 94096325042746502515294076928 

The 3000 through 3002 5-smooth numbers:
278942752080 279936000000 281250000000 

The 3000 through 3002 7-smooth numbers:
50176000 50331648 50388480 

The 3000 through 3002 11-smooth numbers:
2112880 2116800 2117016 

The 3000 through 3002 13-smooth numbers:
390000 390390 390625 

The 3000 through 3002 17-smooth numbers:
145800 145860 146016 

The 3000 through 3002 19-smooth numbers:
74256 74358 74360 

The 3000 through 3002 23-smooth numbers:
46552 46575 46585 

The 3000 through 3002 29-smooth numbers:
33516 33524 33534 

Show twenty numbers starting with 30,000 n-smooth numbers for n=503 through n=521
The 30000 through 30019 503-smooth numbers:
62913 62914 62916 62918 62920 62923 62926 62928 62930 62933 62935 62937 62944 62946 62951 62952 62953 62957 62959 62964 

The 30000 through 30019 509-smooth numbers:
62601 62602 62604 62607 62608 62609 62611 62618 62620 62622 62624 62625 62626 62628 62629 62634 62640 62643 62645 62646 

The 30000 through 30019 521-smooth numbers:
62287 62288 62291 62292 62300 62304 62307 62308 62310 62315 62320 62321 62322 62325 62328 62329 62330 62331 62335 62336 

JavaScript[edit]

function isPrime(n){
    var x = Math.floor(Math.sqrt(n)), i = 2
    
    while ((i <= x) && (n % i != 0)) i++
    
    return (x < i)
}

function smooth(n, s, k){
    var p = []
    for (let i = 2; i <= n; i++){
        if (isPrime(i)){
            p.push([BigInt(i), [1n], 0])
        }
    }
    
    var res = []
    for (let i = 0; i < s + k; i++){
        var m = p[0][1][p[0][2]]
        
        for (let j = 1; j < p.length; j++){
            if (p[j][1][p[j][2]] < m) m = p[j][1][p[j][2]]
        }
        
        for (let j = 0; j < p.length; j++){
            p[j][1].push(p[j][0]*m)
            if (p[j][1][p[j][2]] == m) p[j][2]++
        }
        
        res.push(m)
    }
    
    return res.slice(s-1, s-1+k);
}

// main 

var sOut = ""

for (let x of [[2, 29, 1, 25], [3, 29, 3000, 3], [503, 521, 30000, 20]]){
    for (let n = x[0]; n <= x[1]; n++){
        if (isPrime(n)){
            sOut += x[2] + " to " + (x[2] - 1 + x[3]) + " " + n + "-smooth numbers: " + smooth(n, x[2], x[3]) + "\n"
        }
    }
}

console.log(sOut)
Output:
1 to 25 2-smooth numbers: 1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,65536,131072,262144,524288,1048576,2097152,4194304,8388608,16777216
1 to 25 3-smooth numbers: 1,2,3,4,6,8,9,12,16,18,24,27,32,36,48,54,64,72,81,96,108,128,144,162,192
1 to 25 5-smooth numbers: 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
1 to 25 7-smooth numbers: 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
1 to 25 11-smooth numbers: 1,2,3,4,5,6,7,8,9,10,11,12,14,15,16,18,20,21,22,24,25,27,28,30,32
1 to 25 13-smooth numbers: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,18,20,21,22,24,25,26,27,28
1 to 25 17-smooth numbers: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,20,21,22,24,25,26,27
1 to 25 19-smooth numbers: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,24,25,26
1 to 25 23-smooth numbers: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25
1 to 25 29-smooth numbers: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25
3000 to 3002 3-smooth numbers: 91580367978306252441724649472,92829823186414819915547541504,94096325042746502515294076928
3000 to 3002 5-smooth numbers: 278942752080,279936000000,281250000000
3000 to 3002 7-smooth numbers: 50176000,50331648,50388480
3000 to 3002 11-smooth numbers: 2112880,2116800,2117016
3000 to 3002 13-smooth numbers: 390000,390390,390625
3000 to 3002 17-smooth numbers: 145800,145860,146016
3000 to 3002 19-smooth numbers: 74256,74358,74360
3000 to 3002 23-smooth numbers: 46552,46575,46585
3000 to 3002 29-smooth numbers: 33516,33524,33534
30000 to 30019 503-smooth numbers: 62913,62914,62916,62918,62920,62923,62926,62928,62930,62933,62935,62937,62944,62946,62951,62952,62953,62957,62959,62964
30000 to 30019 509-smooth numbers: 62601,62602,62604,62607,62608,62609,62611,62618,62620,62622,62624,62625,62626,62628,62629,62634,62640,62643,62645,62646
30000 to 30019 521-smooth numbers: 62287,62288,62291,62292,62300,62304,62307,62308,62310,62315,62320,62321,62322,62325,62328,62329,62330,62331,62335,62336

Julia[edit]

using Primes

function nsmooth(N, needed)
    nexts, smooths = [BigInt(i) for i in 2:N if isprime(i)], [BigInt(1)]
    prim, count = deepcopy(nexts), 1
    indices = ones(Int, length(nexts))
    while count < needed
        x = minimum(nexts)
        push!(smooths, x)
        count += 1
        for j in 1:length(nexts)
            (nexts[j] <= x) && (nexts[j] = prim[j] * smooths[(indices[j] += 1)])
        end
    end
    return (smooths[end] > typemax(Int)) ? smooths : Int.(smooths)
end

function testnsmoothfilters()
    for i in filter(isprime, 1:29)
        println("The first 25 n-smooth numbers for n = $i are: ", nsmooth(i, 25))
    end
    for i in filter(isprime, 3:29)
        println("The 3000th through 3002nd ($i)-smooth numbers are: ", nsmooth(i, 3002)[3000:3002])
    end
    for i in filter(isprime, 503:521)
        println("The 30000th through 30019th ($i)-smooth numbers >= 30000 are: ", nsmooth(i, 30019)[30000:30019])
    end
end

testnsmoothfilters()
Output:
The first 25 n-smooth numbers for n = 2 are: [1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216]
The first 25 n-smooth numbers for n = 3 are: [1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192]
The first 25 n-smooth numbers for n = 5 are: [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]
The first 25 n-smooth numbers for n = 7 are: [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]
The first 25 n-smooth numbers for n = 11 are: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32]
The first 25 n-smooth numbers for n = 13 are: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28]
The first 25 n-smooth numbers for n = 17 are: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27]
The first 25 n-smooth numbers for n = 19 are: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26]
The first 25 n-smooth numbers for n = 23 are: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]
The first 25 n-smooth numbers for n = 29 are: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]
The 3000th through 3002nd (3)-smooth numbers are: BigInt[91580367978306252441724649472, 92829823186414819915547541504, 94096325042746502515294076928]
The 3000th through 3002nd (5)-smooth numbers are: [278942752080, 279936000000, 281250000000]
The 3000th through 3002nd (7)-smooth numbers are: [50176000, 50331648, 50388480]
The 3000th through 3002nd (11)-smooth numbers are: [2112880, 2116800, 2117016]
The 3000th through 3002nd (13)-smooth numbers are: [390000, 390390, 390625]
The 3000th through 3002nd (17)-smooth numbers are: [145800, 145860, 146016]
The 3000th through 3002nd (19)-smooth numbers are: [74256, 74358, 74360]
The 3000th through 3002nd (23)-smooth numbers are: [46552, 46575, 46585]
The 3000th through 3002nd (29)-smooth numbers are: [33516, 33524, 33534]
The 30000th through 30019th (503)-smooth numbers >= 30000 are: [62913, 62914, 62916, 62918, 62920, 62923, 62926, 62928, 62930, 62933, 62935, 62937, 62944, 62946, 62951, 62952, 62953, 62957, 62959, 62964]
The 30000th through 30019th (509)-smooth numbers >= 30000 are: [62601, 62602, 62604, 62607, 62608, 62609, 62611, 62618, 62620, 62622, 62624, 62625, 62626, 62628, 62629, 62634, 62640, 62643, 62645, 62646]
The 30000th through 30019th (521)-smooth numbers >= 30000 are: [62287, 62288, 62291, 62292, 62300, 62304, 62307, 62308, 62310, 62315, 62320, 62321, 62322, 62325, 62328, 62329, 62330, 62331, 62335, 62336]

Kotlin[edit]

Translation of: Go
import java.math.BigInteger

var primes = mutableListOf<BigInteger>()
var smallPrimes = mutableListOf<Int>()

// cache all primes up to 521
fun init() {
    val two = BigInteger.valueOf(2)
    val three = BigInteger.valueOf(3)
    val p521 = BigInteger.valueOf(521)
    val p29 = BigInteger.valueOf(29)
    primes.add(two)
    smallPrimes.add(2)
    var i = three
    while (i <= p521) {
        if (i.isProbablePrime(1)) {
            primes.add(i)
            if (i <= p29) {
                smallPrimes.add(i.toInt())
            }
        }
        i += two
    }
}

fun min(bs: List<BigInteger>): BigInteger {
    require(bs.isNotEmpty()) { "slice must have at lease one element" }
    val it = bs.iterator()
    var res = it.next()
    while (it.hasNext()) {
        val t = it.next()
        if (t < res) {
            res = t
        }
    }
    return res
}

fun nSmooth(n: Int, size: Int): List<BigInteger> {
    require(n in 2..521) { "n must be between 2 and 521" }
    require(size >= 1) { "size must be at least 1" }

    val bn = BigInteger.valueOf(n.toLong())
    var ok = false
    for (prime in primes) {
        if (bn == prime) {
            ok = true
            break
        }
    }
    require(ok) { "n must be a prime number" }

    val ns = Array<BigInteger>(size) { BigInteger.ZERO }
    ns[0] = BigInteger.ONE
    val next = mutableListOf<BigInteger>()
    for (i in 0 until primes.size) {
        if (primes[i] > bn) {
            break
        }
        next.add(primes[i])
    }
    val indices = Array(next.size) { 0 }
    for (m in 1 until size) {
        ns[m] = min(next)
        for (i in indices.indices) {
            if (ns[m] == next[i]) {
                indices[i]++
                next[i] = primes[i] * ns[indices[i]]
            }
        }
    }

    return ns.toList()
}

fun main() {
    init()
    for (i in smallPrimes) {
        println("The first 25 $i-smooth numbers are:")
        println(nSmooth(i, 25))
        println()
    }
    for (i in smallPrimes.drop(1)) {
        println("The 3,000th to 3,202 $i-smooth numbers are:")
        println(nSmooth(i, 3_002).drop(2_999))
        println()
    }
    for (i in listOf(503, 509, 521)) {
        println("The 30,000th to 30,019 $i-smooth numbers are:")
        println(nSmooth(i, 30_019).drop(29_999))
        println()
    }
}
Output:
The first 25 2-smooth numbers are:
[1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216]

The first 25 3-smooth numbers are:
[1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192]

The first 25 5-smooth numbers are:
[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]

The first 25 7-smooth numbers are:
[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]

The first 25 11-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32]

The first 25 13-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28]

The first 25 17-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27]

The first 25 19-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26]

The first 25 23-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]

The first 25 29-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]

The 3,000th to 3,202 3-smooth numbers are:
[91580367978306252441724649472, 92829823186414819915547541504, 94096325042746502515294076928]

The 3,000th to 3,202 5-smooth numbers are:
[278942752080, 279936000000, 281250000000]

The 3,000th to 3,202 7-smooth numbers are:
[50176000, 50331648, 50388480]

The 3,000th to 3,202 11-smooth numbers are:
[2112880, 2116800, 2117016]

The 3,000th to 3,202 13-smooth numbers are:
[390000, 390390, 390625]

The 3,000th to 3,202 17-smooth numbers are:
[145800, 145860, 146016]

The 3,000th to 3,202 19-smooth numbers are:
[74256, 74358, 74360]

The 3,000th to 3,202 23-smooth numbers are:
[46552, 46575, 46585]

The 3,000th to 3,202 29-smooth numbers are:
[33516, 33524, 33534]

The 30,000th to 30,019 503-smooth numbers are:
[62913, 62914, 62916, 62918, 62920, 62923, 62926, 62928, 62930, 62933, 62935, 62937, 62944, 62946, 62951, 62952, 62953, 62957, 62959, 62964]

The 30,000th to 30,019 509-smooth numbers are:
[62601, 62602, 62604, 62607, 62608, 62609, 62611, 62618, 62620, 62622, 62624, 62625, 62626, 62628, 62629, 62634, 62640, 62643, 62645, 62646]

The 30,000th to 30,019 521-smooth numbers are:
[62287, 62288, 62291, 62292, 62300, 62304, 62307, 62308, 62310, 62315, 62320, 62321, 62322, 62325, 62328, 62329, 62330, 62331, 62335, 62336]

Mathematica / Wolfram Language[edit]

ClearAll[GenerateSmoothNumbers]
GenerateSmoothNumbers[max_?Positive] := GenerateSmoothNumbers[max, 7]
GenerateSmoothNumbers[max_?Positive, maxprime_?Positive] := 
 Module[{primes, len, vars, body, endspecs, its, data},
  primes = Prime[Range[PrimePi[maxprime]]];
  len = Length[primes];
  If[max < Min[primes],
   {}
   ,
   vars = Table[Unique[], len];
   body = Times @@ (primes^vars);
   endspecs = Prepend[Most[primes^vars], 1];
   endspecs = FoldList[Times, endspecs];
   its = Transpose[{vars, ConstantArray[0, len], MapThread[N@Log[#1, max/#2] &, {primes, endspecs}]}];
   With[{b = body, is = its},
    data = Table[b, Evaluate[Sequence @@ is]];
    ];
   data = Sort[Flatten[data]];
   data
   ]
  ]
Take[GenerateSmoothNumbers[10^8, 2], 25]
Take[GenerateSmoothNumbers[200, 3], 25]
Take[GenerateSmoothNumbers[200, 5], 25]
Take[GenerateSmoothNumbers[200, 7], 25]
Take[GenerateSmoothNumbers[200, 11], 25]
Take[GenerateSmoothNumbers[200, 13], 25]
Take[GenerateSmoothNumbers[200, 17], 25]
Take[GenerateSmoothNumbers[200, 19], 25]
Take[GenerateSmoothNumbers[200, 23], 25]
Take[GenerateSmoothNumbers[200, 29], 25]
Take[GenerateSmoothNumbers[10^40, 3], {3000, 3002}]
Take[GenerateSmoothNumbers[10^15, 5], {3000, 3002}]
Take[GenerateSmoothNumbers[10^10, 7], {3000, 3002}]
Take[GenerateSmoothNumbers[10^7, 11], {3000, 3002}]
Take[GenerateSmoothNumbers[10^7, 13], {3000, 3002}]
Take[GenerateSmoothNumbers[10^6, 17], {3000, 3002}]
Take[GenerateSmoothNumbers[10^5, 19], {3000, 3002}]
Take[GenerateSmoothNumbers[10^5, 23], {3000, 3002}]
Take[GenerateSmoothNumbers[10^5, 29], {3000, 3002}]

s = Select[Range[10^5], FactorInteger /* Last /* First /* LessEqualThan[503]];
s[[30000 ;; 30019]]
s = Select[Range[10^5], FactorInteger /* Last /* First /* LessEqualThan[509]];
s[[30000 ;; 30019]]
s = Select[Range[10^5], FactorInteger /* Last /* First /* LessEqualThan[521]];
s[[30000 ;; 30019]]
Output:
{1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,65536,131072,262144,524288,1048576,2097152,4194304,8388608,16777216}
{1,2,3,4,6,8,9,12,16,18,24,27,32,36,48,54,64,72,81,96,108,128,144,162,192}
{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}
{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}
{1,2,3,4,5,6,7,8,9,10,11,12,14,15,16,18,20,21,22,24,25,27,28,30,32}
{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,18,20,21,22,24,25,26,27,28}
{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,20,21,22,24,25,26,27}
{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,24,25,26}
{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25}
{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25}
{91580367978306252441724649472,92829823186414819915547541504,94096325042746502515294076928}
{278942752080,279936000000,281250000000}
{50176000,50331648,50388480}
{2112880,2116800,2117016}
{390000,390390,390625}
{145800,145860,146016}
{74256,74358,74360}
{46552,46575,46585}
{33516,33524,33534}
{62913,62914,62916,62918,62920,62923,62926,62928,62930,62933,62935,62937,62944,62946,62951,62952,62953,62957,62959,62964}
{62601,62602,62604,62607,62608,62609,62611,62618,62620,62622,62624,62625,62626,62628,62629,62634,62640,62643,62645,62646}
{62287,62288,62291,62292,62300,62304,62307,62308,62310,62315,62320,62321,62322,62325,62328,62329,62330,62331,62335,62336}

Nim[edit]

Translation of: Kotlin
Library: bignum

Translation of Kotlin except for the search of primes where we have chosen to use a sieve of Eratosthenes.

import sequtils, strutils
import bignum

const N = 521


func initPrimes(): tuple[primes: seq[Int]; smallPrimes: seq[int]] =

  var sieve: array[2..N, bool]
  for n in 2..N:
    if not sieve[n]:
      for k in countup(n * n, N, n): sieve[k] = true

  for n, isComposite in sieve:
    if not isComposite:
      result.primes.add newInt(n)
      if n <= 29: result.smallPrimes.add n

# Cache all primes up to N.
let (Primes, SmallPrimes) = initPrimes()



proc nSmooth(n, size: Positive): seq[Int] =
  assert n in 2..N, "'n' must be between 2 and " & $N

  let bn = newInt(n)
  assert bn in Primes, "'n' must be a prime number"

  result.setLen(size)
  result[0] = newInt(1)

  var next: seq[Int]
  for prime in Primes:
    if prime > bn: break
    next.add prime

  var indices = newSeq[int](next.len)
  for m in 1..<size:
    result[m] = next[next.minIndex()]
    for i in 0..indices.high:
      if result[m] == next[i]:
        inc indices[i]
        next[i] = Primes[i] * result[indices[i]]


when isMainModule:

  for n in SmallPrimes:
    echo "The first ", n, "-smooth numbers are:"
    echo nSmooth(n, 25).join(" ")
    echo ""

  for n in SmallPrimes[1..^1]:
    echo "The 3000th to 3202th ", n, "-smooth numbers are:"
    echo nSmooth(n, 3002)[2999..^1].join(" ")
    echo ""

  for n in [503, 509, 521]:
    echo "The 30000th to 30019th ", n, "-smooth numbers are:"
    echo nSmooth(n, 30_019)[29_999..^1].join(" ")
    echo ""
Output:
The first 2-smooth numbers are:
1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576 2097152 4194304 8388608 16777216

The first 3-smooth numbers are:
1 2 3 4 6 8 9 12 16 18 24 27 32 36 48 54 64 72 81 96 108 128 144 162 192

The first 5-smooth numbers are:
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

The first 7-smooth numbers are:
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

The first 11-smooth numbers are:
1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 18 20 21 22 24 25 27 28 30 32

The first 13-smooth numbers are:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 21 22 24 25 26 27 28

The first 17-smooth numbers are:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20 21 22 24 25 26 27

The first 19-smooth numbers are:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 24 25 26

The first 23-smooth numbers are:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

The first 29-smooth numbers are:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

The 3000th to 3202th 3-smooth numbers are:
91580367978306252441724649472 92829823186414819915547541504 94096325042746502515294076928

The 3000th to 3202th 5-smooth numbers are:
278942752080 279936000000 281250000000

The 3000th to 3202th 7-smooth numbers are:
50176000 50331648 50388480

The 3000th to 3202th 11-smooth numbers are:
2112880 2116800 2117016

The 3000th to 3202th 13-smooth numbers are:
390000 390390 390625

The 3000th to 3202th 17-smooth numbers are:
145800 145860 146016

The 3000th to 3202th 19-smooth numbers are:
74256 74358 74360

The 3000th to 3202th 23-smooth numbers are:
46552 46575 46585

The 3000th to 3202th 29-smooth numbers are:
33516 33524 33534

The 30000th to 30019th 503-smooth numbers are:
62913 62914 62916 62918 62920 62923 62926 62928 62930 62933 62935 62937 62944 62946 62951 62952 62953 62957 62959 62964

The 30000th to 30019th 509-smooth numbers are:
62601 62602 62604 62607 62608 62609 62611 62618 62620 62622 62624 62625 62626 62628 62629 62634 62640 62643 62645 62646

The 30000th to 30019th 521-smooth numbers are:
62287 62288 62291 62292 62300 62304 62307 62308 62310 62315 62320 62321 62322 62325 62328 62329 62330 62331 62335 62336

Pascal[edit]

Works with: Extended Pascal
program nSmoothNumbers(output);

const
	primeCount = 538;
	{$ifDef __GPC__} {$setLimit=3889} {$endIf}
	cardinalMax = 3888;

type
	cardinal = 0..cardinalMax;
	cardinals = set of cardinal;
	cardinalPositive = 1..cardinalMax;
	
	natural = 1..maxInt;
	
	primeIndex = 1..primeCount;
	list = array[primeIndex] of cardinal;

var
	prime: list;

{
	\brief performs prime factorization
	
	\param n the number to factorize
	\return list of exponents for a unique integer factorization using primes
	
	This function requires the global `prime` variable to be initialized.
}
function primeFactorization(n: natural): list;
var
	i: primeIndex;
	{
		Instead of writing `[1: 0; 2: 0; 3: 0; …]`, Extended Pascal permits us
		to use an “array completer” for all unspecified values.
	}
	result: list value [otherwise 0];
begin
	for i := 1 to primeCount do
	begin
		while n mod prime[i] = 0 do
		begin
			n := n div prime[i];
			result[i] := result[i] + 1
		end
	end;
	
	{ Nullify result if the utilized `prime` list was insufficient. }
	if n <> 1 then
	begin
		{ `array` literals need an explicit data type specification. }
		result := list[otherwise 0]
	end;
	
	{
		In a Pascal function definition there must be _one_ assignment to the
		(implicitly) declared variable bearing the same name as of the function.
		This will be returned function result.
	}
	primeFactorization := result
end;

{
	\brief returns whether a number is an n-smooth number
	
	\param smoothness the maximum permissible prime factor
	\param x the number to check
	\return whether \param x is smoothness-smooth
}
function isSmooth(
		protected smoothness: cardinalPositive;
		protected x: natural
	): Boolean;
	{
		Return set of non-one factors (i.e. non-zero exponent).
		
		Nesting this function permits most compilers to allocate memory
		for variables only if the function is actually needed.
	}
	function factorizationBases: cardinals;
	var
		exponent: list;
		i: primeIndex;
		result: cardinals value [];
	begin
		exponent := primeFactorization(x);
		
		for i := 1 to primeCount do
		begin
			if exponent[i] > 0 then
			begin
				result := result + [prime[i]]
			end
		end;
		
		factorizationBases := result
	end;
begin
	case smoothness of
		1:
		begin
			{
				The value `1` actually does not have a prime factorization, yet as per
				task specification “1 (unity) is always included in n‑smooth numbers.”
			}
			isSmooth := x = 1
		end;
		{ 2‑smooth numbers is an alias for powers of 2 (OEIS sequence A000079) }
		2:
		begin
			isSmooth := 2 pow round(ln(x) / ln(2)) = x
		end
		{ The `otherwise` catch-all-clause is an Extended Pascal extension. }
		otherwise
		begin
			isSmooth := card(factorizationBases - [0..smoothness]) = 0
			{
				`card` (“cardinality”) is an Extended Pascal extension.
				Testing `card(M) = 0` is equivalent to `M = []` (empty set).
			}
		end	
	end
end;

{ === ancillary routines =============================================== }

{ This is just your regular Eratosthenes prime sieve. }
procedure sieve(var primes: cardinals);
var
	n: natural;
	i: integer;
	multiples: cardinals;
begin
	{ `1` is by definition not a prime number }
	primes := primes - [1];
	
	{ find the next non-crossed number }
	for n := 2 to cardinalMax do
	begin
		if n in primes then
		begin
			multiples := [];
			{ We do _not_ want to remove 1 * n. }
			i := 2 * n;
			while i in [n..cardinalMax] do
			begin
				multiples := multiples + [i];
				i := i + n
			end;
			
			primes := primes - multiples
		end
	end
end;

{ This procedure transforms a `set` to an `array`. }
procedure populatePrime;
var
	primes: cardinals value [1..cardinalMax];
	n: natural value 1;
	i: primeIndex value 1;
begin
	sieve(primes);
	
	for i := 1 to primeCount do
	begin
		repeat
		begin
			n := n + 1
		end
		until n in primes;
		
		prime[i] := n
	end
end;

{ ### MAIN ############################################################# }
var
	i: primeIndex value primeCount;
	candidate: natural;
	count: cardinal;
begin
	populatePrime;
	
	{ Show the first 25 n-smooth numbers […] }
	{ […] only prime [n] are to be used. }
	while prime[i] > 29 do
	begin
		i := i - 1
	end;
	
	{
		In Pascal, `for` loop limits are evaluated _once_,
		so this is perfectly legal:
	}
	for i := 1 to i do
	begin
		write(prime[i]:4, '-smooth: 1');
		
		count := 1; { `1` already listed. }
		candidate := 2;
		
		while count < 25 do
		begin
			if isSmooth(prime[i], candidate) then
			begin
				write(', ', candidate:1);
				count := count + 1
			end;
			candidate := candidate + 1
		end;
		
		writeLn
	end;
	
	{
		Note, in Pascal, the `for`‑loop iteration variable is not modified.
		At this point in the code, `i` (still) has the value after the `while`-loop.
		[This statement is not true, if a `goto` prematurely exited the `for`‑loop.]
	}
	
	writeLn;
	
	{ […] show three numbers starting with 3,000 […] }
	for i := 2 to i do
	begin
		write(prime[i]:4, '-smooth:');
		
		count := 0;
		candidate := 3000;
		
		while count < 3 do
		begin
			if isSmooth(prime[i], candidate) then
			begin
				write(' ', candidate:1);
				count := count + 1
			end;
			candidate := candidate + 1
		end;
		
		writeLn
	end;
end.
Output:
   2-smooth: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216
   3-smooth: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192
   5-smooth: 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
   7-smooth: 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
  11-smooth: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32
  13-smooth: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28
  17-smooth: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27
  19-smooth: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26
  23-smooth: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25
  29-smooth: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25

   3-smooth: 3072 3456 3888
   5-smooth: 3000 3072 3125
   7-smooth: 3000 3024 3072
  11-smooth: 3000 3024 3025
  13-smooth: 3000 3003 3024
  17-smooth: 3000 3003 3024
  19-smooth: 3000 3003 3024
  23-smooth: 3000 3003 3024
  29-smooth: 3000 3003 3016

Perl[edit]

Library: ntheory
use strict;
use warnings;
use feature 'say';
use ntheory qw<primes>;
use List::Util qw<min>;

#use bigint     # works, but slow
use Math::GMPz; # this module gives roughly 16x speed-up

sub smooth_numbers {
#    my(@m) = @_;                               # use with 'bigint'
    my @m = map { Math::GMPz->new($_) } @_;     # comment out to NOT use Math::GMPz
    my @s;
    push @s, [1] for 0..$#m;

    return sub {
    my $n = $s[0][0];
    $n = min $n, $s[$_][0] for 1..$#m;
    for (0..$#m) {
            shift @{$s[$_]} if $s[$_][0] == $n;
            push @{$s[$_]}, $n * $m[$_]
        }
        return $n
    }
}

sub abbrev {
    my($n) = @_;
    return $n if length($n) <= 50;
    substr($n,0,10) . "...(@{[length($n) - 2*10]} digits omitted)..." . substr($n, -10, 10)
}

my @primes = @{primes(10_000)};

my $start = 3000; my $cnt = 3;
for my $n_smooth (0..9) {
    say "\nFirst 25, and ${start}th through @{[$start+2]}nd $primes[$n_smooth]-smooth numbers:";
    my $s = smooth_numbers(@primes[0..$n_smooth]);
    my @S25;
    push @S25, $s->() for 1..25;
    say join ' ', @S25;

    my @Sm; my $c = 25;
    do {
        my $sn = $s->();
        push @Sm, abbrev($sn) if ++$c >= $start;
    } until @Sm == $cnt;
    say join ' ', @Sm;
}

$start = 30000; $cnt = 20;
for my $n_smooth (95..97) { # (503, 509, 521) {
    say "\n${start}th through @{[$start+$cnt-1]}th $primes[$n_smooth]-smooth numbers:";
    my $s = smooth_numbers(@primes[0..$n_smooth]);
    my(@Sm,$c);
    do {
        my $sn = $s->();
        push @Sm, $sn if ++$c >= $start;
    } until @Sm == $cnt;
    say join ' ', @Sm;
}
Output:
First 25, and 3000th through 3002nd 2-smooth numbers:
1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576 2097152 4194304 8388608 16777216
6151159610...(883 digits omitted)...9114994688 1230231922...(884 digits omitted)...8229989376 2460463844...(884 digits omitted)...6459978752

First 25, and 3000th through 3002nd 3-smooth numbers:
1 2 3 4 6 8 9 12 16 18 24 27 32 36 48 54 64 72 81 96 108 128 144 162 192
91580367978306252441724649472 92829823186414819915547541504 94096325042746502515294076928

First 25, and 3000th through 3002nd 5-smooth numbers:
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
278942752080 279936000000 281250000000

First 25, and 3000th through 3002nd 7-smooth numbers:
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
50176000 50331648 50388480

First 25, and 3000th through 3002nd 11-smooth numbers:
1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 18 20 21 22 24 25 27 28 30 32
2112880 2116800 2117016

First 25, and 3000th through 3002nd 13-smooth numbers:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 21 22 24 25 26 27 28
390000 390390 390625

First 25, and 3000th through 3002nd 17-smooth numbers:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20 21 22 24 25 26 27
145800 145860 146016

First 25, and 3000th through 3002nd 19-smooth numbers:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 24 25 26
74256 74358 74360

First 25, and 3000th through 3002nd 23-smooth numbers:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
46552 46575 46585

First 25, and 3000th through 3002nd 29-smooth numbers:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
33516 33524 33534

30000th through 30019th 503-smooth numbers:
62913 62914 62916 62918 62920 62923 62926 62928 62930 62933 62935 62937 62944 62946 62951 62952 62953 62957 62959 62964

30000th through 30019th 509-smooth numbers:
62601 62602 62604 62607 62608 62609 62611 62618 62620 62622 62624 62625 62626 62628 62629 62634 62640 62643 62645 62646

30000th through 30019th 521-smooth numbers:
62287 62288 62291 62292 62300 62304 62307 62308 62310 62315 62320 62321 62322 62325 62328 62329 62330 62331 62335 62336

Phix[edit]

Library: Phix/mpfr
Translation of: Julia
with javascript_semantics 
include mpfr.e
 
function nsmooth(integer n, integer needed)
-- note that n is a prime index, ie 1,2,3,4... for 2,3,5,7...
    sequence smooth = {mpz_init(1)},
             nexts = get_primes(-n),
             indices = repeat(1,n)
    for i=1 to n do nexts[i] = mpz_init(nexts[i]) end for
    for i=2 to needed do
        mpz x = mpz_init_set(mpz_min(nexts))
        smooth = append(smooth,x)
        for j=1 to n do
            if mpz_cmp(nexts[j],x)<=0 then
                indices[j] += 1
                mpz_mul_si(nexts[j],smooth[indices[j]],get_prime(j))
            end if
        end for
    end for
    return smooth
end function
 
function flat_str(sequence s)
    for i=1 to length(s) do s[i] = shorten(mpz_get_str(s[i]),ml:=10) end for
    return join(s," ")
end function
 
for n=1 to 10 do
    printf(1,"%d-smooth[1..25]: %s\n",{get_prime(n),flat_str(nsmooth(n, 25))})
end for
for n=1 to 10 do
    printf(1,"%d-smooth[3000..3002]: %s\n",{get_prime(n),flat_str(nsmooth(n, 3002)[3000..3002])})
end for
for n=96 to 98 do   -- primes 503, 509, and 521
    printf(1,"%d-smooth[30000..30019]: %s\n",{get_prime(n),flat_str(nsmooth(n, 30019)[30000..30019])})
end for
Output:
2-smooth[1..25]: 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576 2097152 4194304 8388608 16777216
3-smooth[1..25]: 1 2 3 4 6 8 9 12 16 18 24 27 32 36 48 54 64 72 81 96 108 128 144 162 192
5-smooth[1..25]: 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
7-smooth[1..25]: 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
11-smooth[1..25]: 1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 18 20 21 22 24 25 27 28 30 32
13-smooth[1..25]: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 21 22 24 25 26 27 28
17-smooth[1..25]: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20 21 22 24 25 26 27
19-smooth[1..25]: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 24 25 26
23-smooth[1..25]: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
29-smooth[1..25]: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
2-smooth[3000..3002]: 615115961...114994688 (903 digits) 123023192...229989376 (904 digits) 246046384...459978752 (904 digits)
3-smooth[3000..3002]: 91580367978306252441724649472 92829823186414819915547541504 94096325042746502515294076928
5-smooth[3000..3002]: 278942752080 279936000000 281250000000
7-smooth[3000..3002]: 50176000 50331648 50388480
11-smooth[3000..3002]: 2112880 2116800 2117016
13-smooth[3000..3002]: 390000 390390 390625
17-smooth[3000..3002]: 145800 145860 146016
19-smooth[3000..3002]: 74256 74358 74360
23-smooth[3000..3002]: 46552 46575 46585
29-smooth[3000..3002]: 33516 33524 33534
503-smooth[30000..30019]: 62913 62914 62916 62918 62920 62923 62926 62928 62930 62933 62935 62937 62944 62946 62951 62952 62953 62957 62959 62964
509-smooth[30000..30019]: 62601 62602 62604 62607 62608 62609 62611 62618 62620 62622 62624 62625 62626 62628 62629 62634 62640 62643 62645 62646
521-smooth[30000..30019]: 62287 62288 62291 62292 62300 62304 62307 62308 62310 62315 62320 62321 62322 62325 62328 62329 62330 62331 62335 62336

Python[edit]

Translation of: C#
primes = [2, 3, 5, 7, 11, 13, 17, 19, 23]

def isPrime(n):
    if n < 2:
        return False

    for i in primes:
        if n == i:
            return True
        if n % i == 0:
            return False
        if i * i > n:
            return True
    print "Oops,", n, " is too large"

def init():
    s = 24
    while s < 600:
        if isPrime(s - 1) and s - 1 > primes[-1]:
            primes.append(s - 1)
        if isPrime(s + 1) and s + 1 > primes[-1]:
            primes.append(s + 1)
        s += 6

def nsmooth(n, size):
    if n < 2 or n > 521:
        raise Exception("n")
    if size < 1:
        raise Exception("n")

    bn = n
    ok = False
    for prime in primes:
        if bn == prime:
            ok = True
            break
    if not ok:
        raise Exception("must be a prime number: n")

    ns = [0] * size
    ns[0] = 1

    next = []
    for prime in primes:
        if prime > bn:
            break
        next.append(prime)

    indicies = [0] * len(next)
    for m in xrange(1, size):
        ns[m] = min(next)
        for i in xrange(0, len(indicies)):
            if ns[m] == next[i]:
                indicies[i] += 1
                next[i] = primes[i] * ns[indicies[i]]

    return ns

def main():
    init()

    for p in primes:
        if p >= 30:
            break
        print "The first", p, "-smooth numbers are:"
        print nsmooth(p, 25)
        print

    for p in primes[1:]:
        if p >= 30:
            break
        print "The 3000 to 3202", p, "-smooth numbers are:"
        print nsmooth(p, 3002)[2999:]
        print

    for p in [503, 509, 521]:
        print "The 30000 to 3019", p, "-smooth numbers are:"
        print nsmooth(p, 30019)[29999:]
        print

main()
Output:
The first 2 -smooth numbers are:
[1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216]

The first 3 -smooth numbers are:
[1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192]

The first 5 -smooth numbers are:
[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]

The first 7 -smooth numbers are:
[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]

The first 11 -smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32]

The first 13 -smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28]

The first 17 -smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27]

The first 19 -smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26]

The first 23 -smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]

The first 29 -smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]

The 3000 to 3202 3 -smooth numbers are:
[91580367978306252441724649472L, 92829823186414819915547541504L, 94096325042746502515294076928L]

The 3000 to 3202 5 -smooth numbers are:
[278942752080L, 279936000000L, 281250000000L]

The 3000 to 3202 7 -smooth numbers are:
[50176000, 50331648, 50388480]

The 3000 to 3202 11 -smooth numbers are:
[2112880, 2116800, 2117016]

The 3000 to 3202 13 -smooth numbers are:
[390000, 390390, 390625]

The 3000 to 3202 17 -smooth numbers are:
[145800, 145860, 146016]

The 3000 to 3202 19 -smooth numbers are:
[74256, 74358, 74360]

The 3000 to 3202 23 -smooth numbers are:
[46552, 46575, 46585]

The 3000 to 3202 29 -smooth numbers are:
[33516, 33524, 33534]

The 30000 to 3019 503 -smooth numbers are:
[62913, 62914, 62916, 62918, 62920, 62923, 62926, 62928, 62930, 62933, 62935, 62937, 62944, 62946, 62951, 62952, 62953, 62957, 62959, 62964]

The 30000 to 3019 509 -smooth numbers are:
[62601, 62602, 62604, 62607, 62608, 62609, 62611, 62618, 62620, 62622, 62624, 62625, 62626, 62628, 62629, 62634, 62640, 62643, 62645, 62646]

The 30000 to 3019 521 -smooth numbers are:
[62287, 62288, 62291, 62292, 62300, 62304, 62307, 62308, 62310, 62315, 62320, 62321, 62322, 62325, 62328, 62329, 62330, 62331, 62335, 62336]

Quackery[edit]

  [ behead 0 swap rot
    witheach 
      [ 2dup < iff
          drop done
        nip nip 
        i^ 1+ swap ] ]               is smallest   ( [ --> n n )

  [ stack ]                          is end.test   (   --> s   )

  [ ]'[ end.test put
    dup temp put
    ' [ 1 ]
    0 rot size of
    [ over end.test share do if done
      [] unrot 
      dup dip
        [ witheach
            [ dip dup peek 
              temp share i^ peek *
              rot swap join swap ] ]
      rot smallest
      dip 
        [ 2dup peek 1+ unrot poke ]
      rot swap over
      -1 peek over = iff
        drop else join
      swap again ]
    drop
    end.test release
    temp release ]                   is smoothwith ( [ --> [   )
    
  [ ' [ 2 3 5 7 11 13 17 19 23 29 ] 
    swap split drop ]                is primes     ( n --> [   )
    
  10 times 
    [ i^ 1+ primes 
      smoothwith [ size 25 = ]
      echo cr ]
  cr
  9 times
    [ i^ 2 + primes
      smoothwith [ size 3002 = ]
      -3 split nip echo cr ]
Output:
[ 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576 2097152 4194304 8388608 16777216 ]
[ 1 2 3 4 6 8 9 12 16 18 24 27 32 36 48 54 64 72 81 96 108 128 144 162 192 ]
[ 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 ]
[ 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 ]
[ 1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 18 20 21 22 24 25 27 28 30 32 ]
[ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 21 22 24 25 26 27 28 ]
[ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20 21 22 24 25 26 27 ]
[ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 24 25 26 ]
[ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 ]
[ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 ]

[ 91580367978306252441724649472 92829823186414819915547541504 94096325042746502515294076928 ]
[ 278942752080 279936000000 281250000000 ]
[ 50176000 50331648 50388480 ]
[ 2112880 2116800 2117016 ]
[ 390000 390390 390625 ]
[ 145800 145860 146016 ]
[ 74256 74358 74360 ]
[ 46552 46575 46585 ]
[ 33516 33524 33534 ]

Raku[edit]

(formerly Perl 6)

Works with: Rakudo version 2019.07.1
sub smooth-numbers (*@list) {
    cache my \Smooth := gather {
        my %i = (flat @list) Z=> (Smooth.iterator for ^@list);
        my %n = (flat @list) Z=> 1 xx *;
 
        loop {
            take my $n := %n{*}.min;
 
            for @list -> \k {
                %n{k} = %i{k}.pull-one * k if %n{k} == $n;
            }
        }
    }
}

sub abbrev ($n) {
   $n.chars > 50 ??
   $n.substr(0,10) ~ "...({$n.chars - 20} digits omitted)..." ~ $n.substr(* - 10) !!
   $n
}
 
my @primes = (2..*).grep: *.is-prime;
 
my $start = 3000;
 
for ^@primes.first( * > 29, :k ) -> $p {
    put join "\n", "\nFirst 25, and {$start}th through {$start+2}nd {@primes[$p]}-smooth numbers:",
    $(smooth-numbers(|@primes[0..$p])[^25]),
    $(smooth-numbers(|@primes[0..$p])[$start - 1 .. $start + 1]».&abbrev);
}
 
$start = 30000;
 
for 503, 509, 521 -> $p {
    my $i = @primes.first( * == $p, :k );
    put "\n{$start}th through {$start+19}th {@primes[$i]}-smooth numbers:\n" ~
    smooth-numbers(|@primes[0..$i])[$start - 1 .. $start + 18];
}
Output:
First 25, and 3000th through 3002nd 2-smooth numbers:
1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576 2097152 4194304 8388608 16777216
6151159610...(883 digits omitted)...9114994688 1230231922...(884 digits omitted)...8229989376 2460463844...(884 digits omitted)...6459978752

First 25, and 3000th through 3002nd 3-smooth numbers:
1 2 3 4 6 8 9 12 16 18 24 27 32 36 48 54 64 72 81 96 108 128 144 162 192
91580367978306252441724649472 92829823186414819915547541504 94096325042746502515294076928

First 25, and 3000th through 3002nd 5-smooth numbers:
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
278942752080 279936000000 281250000000

First 25, and 3000th through 3002nd 7-smooth numbers:
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
50176000 50331648 50388480

First 25, and 3000th through 3002nd 11-smooth numbers:
1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 18 20 21 22 24 25 27 28 30 32
2112880 2116800 2117016

First 25, and 3000th through 3002nd 13-smooth numbers:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 21 22 24 25 26 27 28
390000 390390 390625

First 25, and 3000th through 3002nd 17-smooth numbers:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20 21 22 24 25 26 27
145800 145860 146016

First 25, and 3000th through 3002nd 19-smooth numbers:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 24 25 26
74256 74358 74360

First 25, and 3000th through 3002nd 23-smooth numbers:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
46552 46575 46585

First 25, and 3000th through 3002nd 29-smooth numbers:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
33516 33524 33534

30000th through 30019th 503-smooth numbers:
62913 62914 62916 62918 62920 62923 62926 62928 62930 62933 62935 62937 62944 62946 62951 62952 62953 62957 62959 62964

30000th through 30019th 509-smooth numbers:
62601 62602 62604 62607 62608 62609 62611 62618 62620 62622 62624 62625 62626 62628 62629 62634 62640 62643 62645 62646

30000th through 30019th 521-smooth numbers:
62287 62288 62291 62292 62300 62304 62307 62308 62310 62315 62320 62321 62322 62325 62328 62329 62330 62331 62335 62336

REXX[edit]

/*REXX pgm computes&displays X n-smooth numbers; both X and N can be specified as ranges*/
numeric digits 200                               /*be able to handle some big numbers.  */
parse arg LOx HIx LOn HIn .                      /*obtain optional arguments from the CL*/
if LOx=='' | LOx==","  then LOx=  1              /*Not specified?  Then use the default.*/
if HIx=='' | HIx==","  then HIx= LOx + 24        /* "      "         "   "   "     "    */
if LOn=='' | LOn==","  then LOn=  2              /* "      "         "   "   "     "    */
if HIn=='' | HIn==","  then HIn= LOn + 27        /* "      "         "   "   "     "    */
call genP HIn                                    /*generate enough primes to satisfy HIn*/
@aList= ' a list of the ';              @thru= ' through '  /*literals used with a  SAY.*/

     do j=LOn  to  HIn;  if !.j==0  then iterate /*if not prime, then skip this number. */
     call smooth HIx,j;                 $=       /*invoke SMOOTH; initialize $  (list). */
                     do k=LOx  to HIx;  $= $ #.k /*append a  smooth number to  "  "   " */
                     end   /*k*/
     say center(@aList  th(LOx)  @thru  th(HIx)     ' numbers for' j"-smooth ",  130, "═")
     say strip($);                      say
     end   /*j*/                                 /* [↑]  the $ list has a leading blank.*/
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
genP: procedure expose @. !. #; parse arg x      /*#≡num of primes; @. ≡array of primes.*/
      @.=;      @.1=2; @.2=3; @.3=5; @.4=7; @.5=11; @.6=13; @.7=17; @.8=19; @.9=23;    #=9
      !.=0;     !.2=1; !.3=2; !.5=3; !.7=4; !.11=5; !.13=6; !.17=7; !.19=8; !.23=9
           do k=@.#+6  by 2  until #>=x ;        if k//3==0    then iterate
           parse var  k  ''  -1  _;              if _==5       then iterate
                        do d=4  until @.d**2>k;  if k//@.d==0  then iterate k
                        end   /*d*/
           #= # + 1;    !.k= #;       @.#= k     /*found a prime, bump counter; assign @*/
           end  /*k*/;                return
/*──────────────────────────────────────────────────────────────────────────────────────*/
smooth: procedure expose @. !. #.; parse arg y,p /*obtain the arguments from the invoker*/
        if p==''  then p= 3                      /*Not specified? Then assume Hamming #s*/
        n= !.p                                   /*the number of primes being used.     */
        nn= n - 1;            #.=  0;    #.1= 1  /*an array of n-smooth numbers (so far)*/
        f.=  1                                   /*the indices of factors of a number.  */
                do j=2  for y-1;              _= f.1
                z= @.1 * #._
                             do k=2  for nn;  _= f.k;  v= @.k * #._;    if v<z  then z= v
                             end   /*k*/
                #.j= z
                             do d=1  for n;   _= f.d;  if @.d * #._==z  then f.d= f.d + 1
                             end   /*d*/
                end   /*j*/;                  return
/*──────────────────────────────────────────────────────────────────────────────────────*/
th: parse arg th; return th || word('th st nd rd', 1+(th//10)*(th//100%10\==1)*(th//10<4))
output   when using the default inputs:
════════════════════════════════════ a list of the  1st  through  25th  numbers for 2-smooth ═════════════════════════════════════
1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576 2097152 4194304 8388608 16777216

════════════════════════════════════ a list of the  1st  through  25th  numbers for 3-smooth ═════════════════════════════════════
1 2 3 4 6 8 9 12 16 18 24 27 32 36 48 54 64 72 81 96 108 128 144 162 192

════════════════════════════════════ a list of the  1st  through  25th  numbers for 5-smooth ═════════════════════════════════════
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

════════════════════════════════════ a list of the  1st  through  25th  numbers for 7-smooth ═════════════════════════════════════
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

════════════════════════════════════ a list of the  1st  through  25th  numbers for 11-smooth ════════════════════════════════════
1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 18 20 21 22 24 25 27 28 30 32

════════════════════════════════════ a list of the  1st  through  25th  numbers for 13-smooth ════════════════════════════════════
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 21 22 24 25 26 27 28

════════════════════════════════════ a list of the  1st  through  25th  numbers for 17-smooth ════════════════════════════════════
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20 21 22 24 25 26 27

════════════════════════════════════ a list of the  1st  through  25th  numbers for 19-smooth ════════════════════════════════════
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 24 25 26

════════════════════════════════════ a list of the  1st  through  25th  numbers for 23-smooth ════════════════════════════════════
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

════════════════════════════════════ a list of the  1st  through  25th  numbers for 29-smooth ════════════════════════════════════
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
output   when using the input of:     3000   3002   3   29
══════════════════════════════════ a list of the  3000th  through  3002nd  numbers for 3-smooth ══════════════════════════════════
91580367978306252441724649472 92829823186414819915547541504 94096325042746502515294076928

══════════════════════════════════ a list of the  3000th  through  3002nd  numbers for 5-smooth ══════════════════════════════════
278942752080 279936000000 281250000000

══════════════════════════════════ a list of the  3000th  through  3002nd  numbers for 7-smooth ══════════════════════════════════
50176000 50331648 50388480

═════════════════════════════════ a list of the  3000th  through  3002nd  numbers for 11-smooth ══════════════════════════════════
2112880 2116800 2117016

═════════════════════════════════ a list of the  3000th  through  3002nd  numbers for 13-smooth ══════════════════════════════════
390000 390390 390625

═════════════════════════════════ a list of the  3000th  through  3002nd  numbers for 17-smooth ══════════════════════════════════
145800 145860 146016

═════════════════════════════════ a list of the  3000th  through  3002nd  numbers for 19-smooth ══════════════════════════════════
74256 74358 74360

═════════════════════════════════ a list of the  3000th  through  3002nd  numbers for 23-smooth ══════════════════════════════════
46552 46575 46585

═════════════════════════════════ a list of the  3000th  through  3002nd  numbers for 29-smooth ══════════════════════════════════
33516 33524 33534
output   when using the input of:     30000   30019   503   521
════════════════════════════════ a list of the  30000th  through  30019th  numbers for 503-smooth ════════════════════════════════
62913 62914 62916 62918 62920 62923 62926 62928 62930 62933 62935 62937 62944 62946 62951 62952 62953 62957 62959 62964

════════════════════════════════ a list of the  30000th  through  30019th  numbers for 509-smooth ════════════════════════════════
62601 62602 62604 62607 62608 62609 62611 62618 62620 62622 62624 62625 62626 62628 62629 62634 62640 62643 62645 62646

════════════════════════════════ a list of the  30000th  through  30019th  numbers for 521-smooth ════════════════════════════════
62287 62288 62291 62292 62300 62304 62307 62308 62310 62315 62320 62321 62322 62325 62328 62329 62330 62331 62335 62336

Ruby[edit]

Translation of: Python
def prime?(n)                     # P3 Prime Generator primality test
  return n | 1 == 3 if n < 5      # n: 2,3|true; 0,1,4|false 
  return false if n.gcd(6) != 1   # this filters out 2/3 of all integers
  sqrtN = Integer.sqrt(n)
  pc = -1                         # initial P3 prime candidates value
  until (pc += 6) > sqrtN         # is resgroup 1st prime candidate > sqrtN
    return false if n % pc == 0 || n % (pc + 2) == 0  # if n is composite
  end
  true
end

def gen_primes(a, b)
    (a..b).select { |pc| pc if prime? pc }
end

def nsmooth(n, limit)
    raise "Exception(n or limit)" if n < 2 || n > 521 || limit < 1
    raise "Exception(must be a prime number: n)" unless prime? n
    
    primes = gen_primes(2, n)
    ns = [0] * limit
    ns[0] = 1
    nextp = primes[0..primes.index(n)]

    indices = [0] * nextp.size
    (1...limit).each do |m|
        ns[m] = nextp.min
        (0...indices.size).each do |i|
            if ns[m] == nextp[i]
                indices[i] += 1
                nextp[i] = primes[i] * ns[indices[i]]
            end
        end
    end
    ns
end

gen_primes(2, 29).each do |prime|
    print "The first 25 #{prime}-smooth numbers are: \n"
    print nsmooth(prime, 25)
    puts
end
puts
gen_primes(3, 29).each do |prime|
    print "The 3000 to 3202 #{prime}-smooth numbers are: "
    print nsmooth(prime, 3002)[2999..-1]    # for ruby >= 2.6: (..)[2999..]
    puts
end
puts
gen_primes(503, 521).each do |prime|
    print "The 30,000 to 30,019 #{prime}-smooth numbers are: \n"
    print nsmooth(prime, 30019)[29999..-1]  # for ruby >= 2.6: (..)[29999..]
    puts
end
Output:
The first 25 2-smooth numbers are: 
[1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216]
The first 25 3-smooth numbers are: 
[1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192]
The first 25 5-smooth numbers are: 
[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]
The first 25 7-smooth numbers are: 
[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]
The first 25 11-smooth numbers are: 
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32]
The first 25 13-smooth numbers are: 
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28]
The first 25 17-smooth numbers are: 
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27]
The first 25 19-smooth numbers are: 
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26]
The first 25 23-smooth numbers are: 
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]
The first 25 29-smooth numbers are: 
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]

The 3000 to 3002 3-smooth numbers are: [91580367978306252441724649472, 92829823186414819915547541504, 94096325042746502515294076928]
The 3000 to 3002 5-smooth numbers are: [278942752080, 279936000000, 281250000000]
The 3000 to 3002 7-smooth numbers are: [50176000, 50331648, 50388480]
The 3000 to 3002 11-smooth numbers are: [2112880, 2116800, 2117016]
The 3000 to 3002 13-smooth numbers are: [390000, 390390, 390625]
The 3000 to 3002 17-smooth numbers are: [145800, 145860, 146016]
The 3000 to 3002 19-smooth numbers are: [74256, 74358, 74360]
The 3000 to 3002 23-smooth numbers are: [46552, 46575, 46585]
The 3000 to 3002 29-smooth numbers are: [33516, 33524, 33534]

The 30,000 to 30,019 503-smooth numbers are: 
[62913, 62914, 62916, 62918, 62920, 62923, 62926, 62928, 62930, 62933, 62935, 62937, 62944, 62946, 62951, 62952, 62953, 62957, 62959, 62964]
The 30,000 to 30,019 509-smooth numbers are: 
[62601, 62602, 62604, 62607, 62608, 62609, 62611, 62618, 62620, 62622, 62624, 62625, 62626, 62628, 62629, 62634, 62640, 62643, 62645, 62646]
The 30,000 to 30,019 521-smooth numbers are: 
[62287, 62288, 62291, 62292, 62300, 62304, 62307, 62308, 62310, 62315, 62320, 62321, 62322, 62325, 62328, 62329, 62330, 62331, 62335, 62336]

Rust[edit]

Translation of: C
fn is_prime(n: u32) -> bool {
    if n < 2 {
        return false;
    }
    if n % 2 == 0 {
        return n == 2;
    }
    if n % 3 == 0 {
        return n == 3;
    }
    let mut p = 5;
    while p * p <= n {
        if n % p == 0 {
            return false;
        }
        p += 2;
        if n % p == 0 {
            return false;
        }
        p += 4;
    }
    true
}

fn find_primes(from: u32, to: u32) -> Vec<u32> {
    let mut primes: Vec<u32> = Vec::new();
    for p in from..=to {
        if is_prime(p) {
            primes.push(p);
        }
    }
    primes
}

fn find_nsmooth_numbers(n: u32, count: usize) -> Vec<u128> {
    let primes = find_primes(2, n);
    let num_primes = primes.len();
    let mut result = Vec::with_capacity(count);
    let mut queue = Vec::with_capacity(num_primes);
    let mut index = Vec::with_capacity(num_primes);
    for i in 0..num_primes {
        index.push(0);
        queue.push(primes[i] as u128);
    }
    result.push(1);
    for i in 1..count {
        for p in 0..num_primes {
            if queue[p] == result[i - 1] {
                index[p] += 1;
                queue[p] = result[index[p]] * primes[p] as u128;
            }
        }
        let mut min_index: usize = 0;
        for p in 1..num_primes {
            if queue[min_index] > queue[p] {
                min_index = p;
            }
        }
        result.push(queue[min_index]);
    }
    result
}

fn print_nsmooth_numbers(n: u32, begin: usize, count: usize) {
    let numbers = find_nsmooth_numbers(n, begin + count);
    print!("{}: {}", n, &numbers[begin]);
    for i in 1..count {
        print!(", {}", &numbers[begin + i]);
    }
    println!();
}

fn main() {
    println!("First 25 n-smooth numbers for n = 2 -> 29:");
    for n in 2..=29 {
        if is_prime(n) {
            print_nsmooth_numbers(n, 0, 25);
        }
    }
    println!();
    println!("3 n-smooth numbers starting from 3000th for n = 3 -> 29:");
    for n in 3..=29 {
        if is_prime(n) {
            print_nsmooth_numbers(n, 2999, 3);
        }
    }
    println!();
    println!("20 n-smooth numbers starting from 30,000th for n = 503 -> 521:");
    for n in 503..=521 {
        if is_prime(n) {
            print_nsmooth_numbers(n, 29999, 20);
        }
    }
}
Output:
First 25 n-smooth numbers for n = 2 -> 29:
2: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216
3: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192
5: 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
7: 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
11: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32
13: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28
17: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27
19: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26
23: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25
29: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25

3 n-smooth numbers starting from 3000th for n = 3 -> 29:
3: 91580367978306252441724649472, 92829823186414819915547541504, 94096325042746502515294076928
5: 278942752080, 279936000000, 281250000000
7: 50176000, 50331648, 50388480
11: 2112880, 2116800, 2117016
13: 390000, 390390, 390625
17: 145800, 145860, 146016
19: 74256, 74358, 74360
23: 46552, 46575, 46585
29: 33516, 33524, 33534

20 n-smooth numbers starting from 30,000th for n = 503 -> 521:
503: 62913, 62914, 62916, 62918, 62920, 62923, 62926, 62928, 62930, 62933, 62935, 62937, 62944, 62946, 62951, 62952, 62953, 62957, 62959, 62964
509: 62601, 62602, 62604, 62607, 62608, 62609, 62611, 62618, 62620, 62622, 62624, 62625, 62626, 62628, 62629, 62634, 62640, 62643, 62645, 62646
521: 62287, 62288, 62291, 62292, 62300, 62304, 62307, 62308, 62310, 62315, 62320, 62321, 62322, 62325, 62328, 62329, 62330, 62331, 62335, 62336

Sidef[edit]

func smooth_generator(primes) {
    var s = primes.len.of { [1] }
    {
        var n = s.map { .first }.min
        { |i|
            s[i].shift if (s[i][0] == n)
            s[i] << (n * primes[i])
        } * primes.len
        n
    }
}

for p in (primes(2,29)) {
    var g = smooth_generator(p.primes)
    say ("First 25 #{'%2d'%p}-smooth numbers: ", 25.of { g.run }.join(' '))
}

say ''

for p in (primes(3,29)) {
    var g = smooth_generator(p.primes)
    say ("3,000th through 3,002nd #{'%2d'%p}-smooth numbers: ", 3002.of { g.run }.last(3).join(' '))
}
Output:
First 25  2-smooth numbers: 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576 2097152 4194304 8388608 16777216
First 25  3-smooth numbers: 1 2 3 4 6 8 9 12 16 18 24 27 32 36 48 54 64 72 81 96 108 128 144 162 192
First 25  5-smooth numbers: 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
First 25  7-smooth numbers: 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
First 25 11-smooth numbers: 1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 18 20 21 22 24 25 27 28 30 32
First 25 13-smooth numbers: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 21 22 24 25 26 27 28
First 25 17-smooth numbers: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20 21 22 24 25 26 27
First 25 19-smooth numbers: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 24 25 26
First 25 23-smooth numbers: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
First 25 29-smooth numbers: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

3,000th through 3,002nd  3-smooth numbers: 91580367978306252441724649472 92829823186414819915547541504 94096325042746502515294076928
3,000th through 3,002nd  5-smooth numbers: 278942752080 279936000000 281250000000
3,000th through 3,002nd  7-smooth numbers: 50176000 50331648 50388480
3,000th through 3,002nd 11-smooth numbers: 2112880 2116800 2117016
3,000th through 3,002nd 13-smooth numbers: 390000 390390 390625
3,000th through 3,002nd 17-smooth numbers: 145800 145860 146016
3,000th through 3,002nd 19-smooth numbers: 74256 74358 74360
3,000th through 3,002nd 23-smooth numbers: 46552 46575 46585
3,000th through 3,002nd 29-smooth numbers: 33516 33524 33534

Optionally, an efficient algorithm for checking if a given arbitrary large number is smooth over a given product of primes:

func is_smooth_over_prod(n, k) {

    return true  if (n == 1)
    return false if (n <= 0)

    for (var g = gcd(n,k); g > 1; g = gcd(n,k)) {
        n /= g**valuation(n,g)        # remove any divisibility by g
        return true if (n == 1)       # smooth if n == 1
    }

    return false
}

for p in (503, 509, 521) {
    var k = p.primorial
    var a = {|n| is_smooth_over_prod(n, k) }.first(30_019).last(20)
    say ("30,000th through 30,019th #{p}-smooth numbers: ", a.join(' '))
}
Output:
30,000th through 30,019th 503-smooth numbers: 62913 62914 62916 62918 62920 62923 62926 62928 62930 62933 62935 62937 62944 62946 62951 62952 62953 62957 62959 62964
30,000th through 30,019th 509-smooth numbers: 62601 62602 62604 62607 62608 62609 62611 62618 62620 62622 62624 62625 62626 62628 62629 62634 62640 62643 62645 62646
30,000th through 30,019th 521-smooth numbers: 62287 62288 62291 62292 62300 62304 62307 62308 62310 62315 62320 62321 62322 62325 62328 62329 62330 62331 62335 62336

Swift[edit]

import BigInt
import Foundation

extension BinaryInteger {
  @inlinable
  public var isPrime: Bool {
    if self == 0 || self == 1 {
      return false
    } else if self == 2 {
      return true
    }

    let max = Self(ceil((Double(self).squareRoot())))

    for i in stride(from: 2, through: max, by: 1) {
      if self % i == 0 {
        return false
      }
    }

    return true
  }
}

@inlinable
public func smoothN<T: BinaryInteger>(n: T, count: Int) -> [T] {
  let primes = stride(from: 2, to: n + 1, by: 1).filter({ $0.isPrime })
  var next = primes
  var indices = [Int](repeating: 0, count: primes.count)
  var res = [T](repeating: 0, count: count)

  res[0] = 1

  guard count > 1 else {
    return res
  }

  for m in 1..<count {
    res[m] = next.min()!

    for i in 0..<indices.count where res[m] == next[i] {
      indices[i] += 1
      next[i] = primes[i] * res[indices[i]]
    }
  }

  return res
}

for n in 2...29 where n.isPrime {
  print("The first 25 \(n)-smooth numbers are: \(smoothN(n: n, count: 25))")
}

print()

for n in 3...29 where n.isPrime {
  print("The 3000...3002 \(n)-smooth numbers are: \(smoothN(n: BigInt(n), count: 3002).dropFirst(2999).prefix(3))")
}

print()

for n in 503...521 where n.isPrime {
  print("The 30,000...30,019 \(n)-smooth numbers are: \(smoothN(n: BigInt(n), count: 30_019).dropFirst(29999).prefix(20))")
}
Output:
The first 25 2-smooth numbers are: [1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216]
The first 25 3-smooth numbers are: [1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192]
The first 25 5-smooth numbers are: [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]
The first 25 7-smooth numbers are: [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]
The first 25 11-smooth numbers are: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32]
The first 25 13-smooth numbers are: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28]
The first 25 17-smooth numbers are: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27]
The first 25 19-smooth numbers are: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26]
The first 25 23-smooth numbers are: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]
The first 25 29-smooth numbers are: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]

The 3000...3002 3-smooth numbers are: [91580367978306252441724649472, 92829823186414819915547541504, 94096325042746502515294076928]
The 3000...3002 5-smooth numbers are: [278942752080, 279936000000, 281250000000]
The 3000...3002 7-smooth numbers are: [50176000, 50331648, 50388480]
The 3000...3002 11-smooth numbers are: [2112880, 2116800, 2117016]
The 3000...3002 13-smooth numbers are: [390000, 390390, 390625]
The 3000...3002 17-smooth numbers are: [145800, 145860, 146016]
The 3000...3002 19-smooth numbers are: [74256, 74358, 74360]
The 3000...3002 23-smooth numbers are: [46552, 46575, 46585]
The 3000...3002 29-smooth numbers are: [33516, 33524, 33534]

The 30,000...30,019 503-smooth numbers are: [62913, 62914, 62916, 62918, 62920, 62923, 62926, 62928, 62930, 62933, 62935, 62937, 62944, 62946, 62951, 62952, 62953, 62957, 62959, 62964]
The 30,000...30,019 509-smooth numbers are: [62601, 62602, 62604, 62607, 62608, 62609, 62611, 62618, 62620, 62622, 62624, 62625, 62626, 62628, 62629, 62634, 62640, 62643, 62645, 62646]
The 30,000...30,019 521-smooth numbers are: [62287, 62288, 62291, 62292, 62300, 62304, 62307, 62308, 62310, 62315, 62320, 62321, 62322, 62325, 62328, 62329, 62330, 62331, 62335, 62336]

Wren[edit]

Translation of: Go
Library: Wren-math
Library: Wren-big
import "/math" for Int
import "/big" for BigInt, BigInts

// cache all primes up to 521
var smallPrimes = Int.primeSieve(521)
var primes = smallPrimes.map { |p| BigInt.new(p) }.toList

var nSmooth = Fn.new { |n, size|
    if (n < 2 || n > 521) Fiber.abort("n must be between 2 and 521")
    if (size < 1) Fiber.abort("size must be at least 1")
    var bn = BigInt.new(n)
    var ok = false
    for (prime in primes) {
        if (bn == prime) {
            ok = true
            break
        }
    }
    if (!ok) Fiber.abort("n must be a prime number")
    var ns = List.filled(size, null)
    ns[0] = BigInt.one
    var next = []
    for (i in 0...primes.count) {
        if (primes[i] > bn) break
        next.add(primes[i])
    }
    var indices = List.filled(next.count, 0)
    for (m in 1...size) {
        ns[m] = BigInts.min(next)
        for (i in 0...indices.count) {
            if (ns[m] == next[i]) {
                indices[i] = indices[i] + 1
                next[i] = primes[i] * ns[indices[i]]
            }
        }
    }
    return ns
}

smallPrimes = smallPrimes.where { |p| p <= 29 }
for (i in smallPrimes) {
    System.print("The first 25 %(i)-smooth numbers are:")
    System.print(nSmooth.call(i, 25))
    System.print()
}
for (i in smallPrimes.skip(1)) {
    System.print("The 3,000th to 3,202nd %(i)-smooth numbers are:")
    System.print(nSmooth.call(i, 3002)[2999..-1])
    System.print()
}
for (i in [503, 509, 521]) {
    System.print("The 30,000th to 30,019th %(i)-smooth numbers are:")
    System.print(nSmooth.call(i, 30019)[29999..-1])
    System.print()
}
Output:
The first 25 2-smooth numbers are:
[1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216]

The first 25 3-smooth numbers are:
[1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192]

The first 25 5-smooth numbers are:
[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]

The first 25 7-smooth numbers are:
[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]

The first 25 11-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32]

The first 25 13-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28]

The first 25 17-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27]

The first 25 19-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26]

The first 25 23-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]

The first 25 29-smooth numbers are:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]

The 3,000th to 3,202nd 3-smooth numbers are:
[91580367978306252441724649472, 92829823186414819915547541504, 94096325042746502515294076928]

The 3,000th to 3,202nd 5-smooth numbers are:
[278942752080, 279936000000, 281250000000]

The 3,000th to 3,202nd 7-smooth numbers are:
[50176000, 50331648, 50388480]

The 3,000th to 3,202nd 11-smooth numbers are:
[2112880, 2116800, 2117016]

The 3,000th to 3,202nd 13-smooth numbers are:
[390000, 390390, 390625]

The 3,000th to 3,202nd 17-smooth numbers are:
[145800, 145860, 146016]

The 3,000th to 3,202nd 19-smooth numbers are:
[74256, 74358, 74360]

The 3,000th to 3,202nd 23-smooth numbers are:
[46552, 46575, 46585]

The 3,000th to 3,202nd 29-smooth numbers are:
[33516, 33524, 33534]

The 30,000th to 30,019th 503-smooth numbers are:
[62913, 62914, 62916, 62918, 62920, 62923, 62926, 62928, 62930, 62933, 62935, 62937, 62944, 62946, 62951, 62952, 62953, 62957, 62959, 62964]

The 30,000th to 30,019th 509-smooth numbers are:
[62601, 62602, 62604, 62607, 62608, 62609, 62611, 62618, 62620, 62622, 62624, 62625, 62626, 62628, 62629, 62634, 62640, 62643, 62645, 62646]

The 30,000th to 30,019th 521-smooth numbers are:
[62287, 62288, 62291, 62292, 62300, 62304, 62307, 62308, 62310, 62315, 62320, 62321, 62322, 62325, 62328, 62329, 62330, 62331, 62335, 62336]

zkl[edit]

Translation of: Go
Library: GMP
GNU Multiple Precision Arithmetic Library and primes
var [const] BI=Import("zklBigNum");  // libGMP

fcn nSmooth(n,sz){	// --> List of big ints
   if(sz<1) throw(Exception.ValueError("size must be at least 1"));
   bn,primes,ns := BI(n), List(), List.createLong(sz);
   if(not bn.probablyPrime()) throw(Exception.ValueError("n must be prime"));
   p:=BI(1); while(p<n){ primes.append(p.nextPrime().copy()) }  // includes n
   ns.append(BI(1));
   next:=primes.copy();
   if(Void!=( z:=primes.find(bn)) ) next.del(z+1,*);

   indices:=List.createLong(next.len(),0);
   do(sz-1){
      ns.append( nm:=BI( next.reduce(fcn(a,b){ a.min(b) }) ));
      foreach i in (indices.len()){
         if(nm==next[i]){
	    indices[i]+=1;
	    next[i]=primes[i]*ns[indices[i]];
	 }
      }
   }
   ns
}
smallPrimes:=List();
p:=BI(1); while(p<29) { smallPrimes.append(p.nextPrime().toInt()) }

foreach p in (smallPrimes){
   println("The first 25 %d-smooth numbers are:".fmt(p));
   println(nSmooth(p,25).concat(" "), "\n")
}
foreach p in (smallPrimes[1,*]){
   print("The 3,000th to 3,202nd %d-smooth numbers are: ".fmt(p));
   println(nSmooth(p,3002)[2999,*].concat(" "));
}
foreach p in (T(503,509,521)){
   println("\nThe 30,000th to 30,019th %d-smooth numbers are:".fmt(p));
   println(nSmooth(p,30019)[29999,*].concat(" "));
}
Output:
The first 25 2-smooth numbers are:
1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576 2097152 4194304 8388608 16777216

The first 25 3-smooth numbers are:
1 2 3 4 6 8 9 12 16 18 24 27 32 36 48 54 64 72 81 96 108 128 144 162 192

The first 25 5-smooth numbers are:
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

The first 25 7-smooth numbers are:
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

The first 25 11-smooth numbers are:
1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 18 20 21 22 24 25 27 28 30 32

The first 25 13-smooth numbers are:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 21 22 24 25 26 27 28

The first 25 17-smooth numbers are:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20 21 22 24 25 26 27

The first 25 19-smooth numbers are:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 24 25 26

The first 25 23-smooth numbers are:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

The first 25 29-smooth numbers are:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

The 3,000th to 3,202nd 3-smooth numbers are: 91580367978306252441724649472 92829823186414819915547541504 94096325042746502515294076928
The 3,000th to 3,202nd 5-smooth numbers are: 278942752080 279936000000 281250000000
The 3,000th to 3,202nd 7-smooth numbers are: 50176000 50331648 50388480
The 3,000th to 3,202nd 11-smooth numbers are: 2112880 2116800 2117016
The 3,000th to 3,202nd 13-smooth numbers are: 390000 390390 390625
The 3,000th to 3,202nd 17-smooth numbers are: 145800 145860 146016
The 3,000th to 3,202nd 19-smooth numbers are: 74256 74358 74360
The 3,000th to 3,202nd 23-smooth numbers are: 46552 46575 46585
The 3,000th to 3,202nd 29-smooth numbers are: 33516 33524 33534

The 30,000th to 30,019th 503-smooth numbers are:
62913 62914 62916 62918 62920 62923 62926 62928 62930 62933 62935 62937 62944 62946 62951 62952 62953 62957 62959 62964

The 30,000th to 30,019th 509-smooth numbers are:
62601 62602 62604 62607 62608 62609 62611 62618 62620 62622 62624 62625 62626 62628 62629 62634 62640 62643 62645 62646

The 30,000th to 30,019th 521-smooth numbers are:
62287 62288 62291 62292 62300 62304 62307 62308 62310 62315 62320 62321 62322 62325 62328 62329 62330 62331 62335 62336