# 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   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
```

•   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.

References

## 11l

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 = 1

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

V indicies =  * 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

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, 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++

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 = 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#

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() {

BigInteger i = 3;
while (i <= 521) {
if (IsPrime(i)) {
if (i <= 29) {
}
}
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 = 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;
}
}

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

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 = 1.to_big_i
nextp = primes[0..primes.index(n)].map { |prm| prm.to_big_i }

indices =  * 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

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 = 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

Thanks for Rudy Velthuis
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;
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');

for i := 1 to size - 1 do

next := TList<BigInteger>.Create;

for p in primes do
begin
if p > bn then
Break;
end;

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

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;

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

if i <= 29 then
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;
end.
```

## Elm

As mentioned on [Hamming numbers task page], currently Elm has many restrictions that make it difficult to implement finding a sequence of these numbers; as N-smooth numbers are just a more general case of Hamming numbers which include the latter, the code from that page is just re-factored to cover the more general case as follows:

```module Main exposing ( main )

import Bitwise exposing (..)
import BigInt exposing ( BigInt )
import Html exposing ( div, text, br )
import Browser exposing ( element )
import Time exposing ( now, posixToMillis )

-- an infinite non-empty non-memoizing Co-Inductive Stream (CIS)...
type CIS a = CIS a (() -> CIS a)

takeCIS2String : Int -> (a -> String) -> CIS a -> String
takeCIS2String n cnvf (CIS ohd otlf) =
let loop i (CIS hd tl) str =
if i < 1 then str
else loop (i - 1) (tl()) (str ++ ", " ++ cnvf hd)
in loop (n - 1) (otlf()) (cnvf ohd)

dropCIS : Int -> CIS a -> CIS a
dropCIS n (CIS _ tl as cis) =
if n < 1 then cis else dropCIS (n - 1) (tl())

-- Priority Queue definition...
type PriorityQ comparable v =
Mt
| Br comparable v (PriorityQ comparable v)
(PriorityQ comparable v)

emptyPQ : PriorityQ comparable v
emptyPQ = Mt

peekMinPQ : PriorityQ comparable v -> Maybe (comparable, v)
peekMinPQ  pq = case pq of
(Br k v _ _) -> Just (k, v)
Mt -> Nothing

pushPQ : comparable -> v -> PriorityQ comparable v
-> PriorityQ comparable v
pushPQ wk wv pq =
case pq of
Mt -> Br wk wv Mt Mt
(Br vk vv pl pr) ->
if wk <= vk then Br wk wv (pushPQ vk vv pr) pl
else Br vk vv (pushPQ wk wv pr) pl

siftdown : comparable -> v -> PriorityQ comparable v
-> PriorityQ comparable v -> PriorityQ comparable v
siftdown wk wv pql pqr =
case pql of
Mt -> Br wk wv Mt Mt
(Br vkl vvl pll prl) ->
case pqr of
Mt -> if wk <= vkl then Br wk wv pql Mt
else Br vkl vvl (Br wk wv Mt Mt) Mt
(Br vkr vvr plr prr) ->
if wk <= vkl && wk <= vkr then Br wk wv pql pqr
else if vkl <= vkr then Br vkl vvl (siftdown wk wv pll prl) pqr
else Br vkr vvr pql (siftdown wk wv plr prr)

replaceMinPQ : comparable -> v -> PriorityQ comparable v
-> PriorityQ comparable v
replaceMinPQ wk wv pq = case pq of
Mt -> Mt
(Br _ _ pl pr) -> siftdown wk wv pl pr

primesTo : Int -> List Int
primesTo n =
if n < 3 then if n < 2 then [] else  else
let oddPrimesTo on =
let sqrtlmt = toFloat on |> sqrt |> truncate
obps = if sqrtlmt < 3 then [] else oddPrimesTo sqrtlmt
ns = List.range 0 ((on - 3) // 2) -- [ 3 .. 2 .. on ]
|> List.map ((+) 3 << (*) 2)
filtfnc fn = List.all (\ bp -> bp * bp > fn ||
modBy bp fn /= 0) obps
in List.filter filtfnc ns
in 2 :: oddPrimesTo n

smooths : Int -> CIS BigInt
smooths n =
let infcis v = CIS v <| \ _ -> infcis (BigInt.add v (BigInt.fromInt 1))
dflt = (0.0, BigInt.fromInt 1) in
if n < 2 then infcis (BigInt.fromInt 1) else
let prms = primesTo n |> List.reverse
|> List.map (\ p -> (logBase 2 (toFloat p), BigInt.fromInt p))
((lgfrstp, frstp) as frstpr) = List.head prms |> Maybe.withDefault dflt
rstps = List.tail prms |> Maybe.withDefault []
frstcis =
let nxt ((lg, v) as vpr) =
CIS vpr <| \ _ -> nxt (lg + lgfrstp, BigInt.mul v frstp)
in nxt frstpr
mkcis ((lg, p) as pr) cis =
let nxt pq (CIS ((lghd, hd) as hdpr) tlf as cs) =
let ((lgv, v) as vpr) = peekMinPQ pq |> Maybe.withDefault dflt in
if BigInt.lt v hd then CIS vpr <| \ _ ->
nxt (replaceMinPQ (lgv + lg) (BigInt.mul v p) pq) cs
else CIS hdpr <| \ _ ->
nxt (pushPQ (lghd + lg) (BigInt.mul hd p) pq) (tlf())
in CIS pr <| \ _ -> nxt (pushPQ (lg + lg) (BigInt.mul p p) emptyPQ) cis
rest() = List.foldl mkcis frstcis rstps
unpr (CIS (_, hd) tlf) = CIS hd <| \ _ -> unpr (tlf())
in CIS (BigInt.fromInt 1) <| \ _ -> unpr (rest())

timemillis : () -> Task Never Int -- a side effect function
timemillis() = now |> andThen (\ t -> succeed (posixToMillis t))

test : () -> Cmd Msg -- side effect function chain (includes "perform")...
test() =
timemillis()
|> andThen (\ strt ->
let test1 = primesTo 29 |> List.map ( \ p ->
[ "The first 25 " ++ String.fromInt p ++ "-smooths:"
, smooths p |> takeCIS2String 25 BigInt.toString
, "" ])
test2 = primesTo 29 |> List.drop 1 |> List.map ( \ p ->
[ "The first three from the 3,000th "
++ String.fromInt p ++ "-smooth numbers are:"
, smooths p |> dropCIS 2999
|> takeCIS2String 3 BigInt.toString
, "" ])
test3 = primesTo 521 |> List.filter ((<=) 503) |> List.map ( \ p ->
[ "The first 20 30,000th up "
++ String.fromInt p ++ "-smooth numbers are:"
, smooths p |> dropCIS 29999
|> takeCIS2String 20 BigInt.toString
, "" ])
in timemillis()
|> andThen (\ stop ->
succeed ([test1, test2, test3, [[ "This took "
++ String.fromInt (stop - strt)
++ " milliseconds."]]]
|> List.concat |> List.concat)))
|> perform Done

-- following code has to do with outputting to a web page using MUV/TEA...
type alias Model = List String

type Msg = Done Model

main : Program () Model Msg
main = -- starts with empty list of strings; views model of filled list...
element { init = \ _ -> ( [], test() )
, update = \ (Done mdl) _ -> ( mdl , Cmd.none )
, subscriptions = \ _ -> Sub.none
, view = \ mdl ->
div [] <| List.map (\ s ->
if s == "" then br [] []
else div [] <| List.singleton <| text s) mdl
}
```
Output:
```The first 25 13-smooths:
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-smooths:
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-smooths:
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-smooths:
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-smooths:
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 three from the 3,000th 3-smooth numbers are:
91580367978306252441724649472, 92829823186414819915547541504, 94096325042746502515294076928

The first three from the 3,000th 5-smooth numbers are:
278942752080, 279936000000, 281250000000

The first three from the 3,000th 7-smooth numbers are:
50176000, 50331648, 50388480

The first three from the 3,000th 11-smooth numbers are:
2112880, 2116800, 2117016

The first three from the 3,000th 13-smooth numbers are:
390000, 390390, 390625

The first three from the 3,000th 17-smooth numbers are:
145800, 145860, 146016

The first three from the 3,000th 19-smooth numbers are:
74256, 74358, 74360

The first three from the 3,000th 23-smooth numbers are:
46552, 46575, 46585

The first three from the 3,000th 29-smooth numbers are:
33516, 33524, 33534

The first 20 30,000th up 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 first 20 30,000th up 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 first 20 30,000th up 521-smooth numbers are:
62287, 62288, 62291, 62292, 62300, 62304, 62307, 62308, 62310, 62315, 62320, 62321, 62322, 62325, 62328, 62329, 62330, 62331, 62335, 62336

This took 506 milliseconds.```

## F#

The easy way to solve this is just to translate the Haskell contribution as from "Hamming Numbers without duplicates", along with a version of a trial division small primes determination since the required primes have such a small range, as follows:

```let primesTo n =
if n < 3 then (if n < 2 then Seq.empty else Seq.singleton 2) else
let rec oddPrimesTo on =
let sqrtlmt = double on |> sqrt |> truncate |> int
let obps = if sqrtlmt < 3 then Seq.empty else oddPrimesTo sqrtlmt
let ns = [ 3 .. 2 .. on ]
let filtfnc fn = Seq.forall (fun bp -> bp * bp > fn ||
fn % bp <> 0) obps
Seq.filter filtfnc ns
Seq.append  (Seq.singleton 2)  (oddPrimesTo n)

type LazyList<'a> = Cons of 'a * Lazy<LazyList<'a>>

// Doesn't need to be that efficient for the task...
#nowarn "40" // don't need to warn for recursive values
let smooths p =
if p < 2 then Seq.singleton (bigint 1) else
let smthprms = primesTo p |> Seq.rev |> Seq.map bigint
let rstps = Seq.tail smthprms
let frstll =
let rec nxt n =
Cons(n, lazy nxt (n * frstp))
nxt frstp
let smult m lzylst =
let rec smlt (Cons(x, rxs)) =
Cons(m * x, lazy(smlt (rxs.Force())))
smlt lzylst
let rec merge (Cons(x, f) as xs) (Cons(y, g) as ys) =
if x < y then Cons(x, lazy(merge (f.Force()) ys))
else Cons(y, lazy(merge xs (g.Force())))
let u s n =
let rec r = merge s (smult n (Cons(1I, lazy r))) in r
Seq.unfold (fun (Cons(hd, rst)) -> Some (hd, rst.Value))
(Cons(1I, lazy(Seq.fold u frstll rstps)))

let strt = System.DateTime.Now.Ticks

primesTo 29 |> Seq.iter (fun p ->
printfn "First 25 %d-smooth:" p
smooths p |> Seq.take 25 |> Seq.toList |> printfn "%A\r\n")

primesTo 29 |> Seq.skip 1 |> Seq.iter (fun p ->
printfn "The first three from the 3,000th %d-smooth numbers are:" p
smooths p |> Seq.skip 2999 |> Seq.take 3 |> Seq.toList |> printfn "%A\r\n")

primesTo 521 |> Seq.skipWhile ((>) 503) |> Seq.iter (fun p ->
printfn "The first 20 30,000th up %d-smooth numbers are:" p
smooths p |> Seq.skip 29999 |> Seq.take 20 |> Seq.toList |> printfn "%A\r\n")

let stop = System.DateTime.Now.Ticks
printfn "This took %d milliseconds." ((stop - strt) / 10000L)
```
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 first three from the 3,000th 3-smooth numbers are:
[91580367978306252441724649472; 92829823186414819915547541504;
94096325042746502515294076928]

The first three from the 3,000th 5-smooth numbers are:
[278942752080; 279936000000; 281250000000]

The first three from the 3,000th 7-smooth numbers are:
[50176000; 50331648; 50388480]

The first three from the 3,000th 11-smooth numbers are:
[2112880; 2116800; 2117016]

The first three from the 3,000th 13-smooth numbers are:
[390000; 390390; 390625]

The first three from the 3,000th 17-smooth numbers are:
[145800; 145860; 146016]

The first three from the 3,000th 19-smooth numbers are:
[74256; 74358; 74360]

The first three from the 3,000th 23-smooth numbers are:
[46552; 46575; 46585]

The first three from the 3,000th 29-smooth numbers are:
[33516; 33524; 33534]

The first 20 30,000th up 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 first 20 30,000th up 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 first 20 30,000th up 521-smooth numbers are:
[62287; 62288; 62291; 62292; 62300; 62304; 62307; 62308; 62310; 62315; 62320;
62321; 62322; 62325; 62328; 62329; 62330; 62331; 62335; 62336]

This took 544 milliseconds.```

As run on an Intel i5-6500 (3.6 GHz boosted when single-threaded), this isn't particularly fast, slowed by DotNet's poor allocation/deallocation of small memory area performance as required here for the "LazyList" implementation (a new allocation for each element) as well as the time to process the deferred execution "thunks" required for memoization, and also because the "BigInt" implementation isn't likely as fast as the "native" implementation used by some languages such as Haskell (by default).

Faster Non-recursive Version

The following code is over twice as fast because it no longer requires a memoized "LazyList" but just the deferred-execution tails of a Co-Inductive Stream (CIS), although to be general it still uses these streams for each merged CIS of the accumulated intermediate result streams, meaning that it still uses many allocations/deallocations for each CIS element; as well, it avoids some of the slow F# "Seq" operations by directly manipulating the "CIS" streams:

```let primesTo n =
if n < 3 then (if n < 2 then Seq.empty else Seq.singleton 2) else
let rec oddPrimesTo on =
let sqrtlmt = double on |> sqrt |> truncate |> int
let obps = if sqrtlmt < 3 then Seq.empty else oddPrimesTo sqrtlmt
let ns = [ 3 .. 2 .. on ]
let filtfnc fn = Seq.forall (fun bp -> bp * bp > fn ||
fn % bp <> 0) obps
Seq.filter filtfnc ns
Seq.append  (Seq.singleton 2)  (oddPrimesTo n)

type CIS<'a> = CIS of 'a * (Unit -> CIS<'a>)

let rec skipCIS n (CIS(_, tlf) as cis) =
if n <= 0 then cis else skipCIS (n - 1) (tlf())

let stringCIS n (CIS(fhd, ftlf)) =
let rec addstr i (CIS(hd, tlf)) str =
if i <= 0 then str + " )"
else addstr (i - 1) (tlf()) (str + ", " + string hd)
addstr (n - 1) (ftlf()) ("( " + string fhd)

type Deque<'a> = Deque of int * int * int * 'a array

let makeDQ v =
let arr = Array.zeroCreate 1024 in arr. <- v
Deque(1023, 0, 1, arr)

let growDQ (Deque(msk, hdi, tli, arr)) =
let sz = arr.Length
let nsz = if sz = 0 then 1024 else sz + sz
let narr =  Array.zeroCreate nsz
let nhdi, ntli =
if hdi = 0 then Array.blit arr 0 narr 0 sz
hdi, sz
else let mv = hdi + sz // move top queue up...
Array.blit arr 0 narr 0 tli
Array.blit arr hdi narr mv (sz - hdi)
mv, tli
Deque(nsz - 1, nhdi, ntli, narr)

let pushDQ v (Deque(_, hdi, tli, _) as dq) =
let (Deque(nmsk, nhdi, ntli, narr)) = if tli <> hdi then dq
else growDQ dq
narr.[ntli] <- v
Deque(nmsk, nhdi, (ntli + 1) &&& nmsk, narr)

// Deque is never empty after the first push and always push before pull!
let inline peekDQ (Deque(_, hdi, _, arr)) = arr.[hdi]
let pullDQ (Deque(msk, hdi, tli, arr)) =
Deque(msk, (hdi + 1) &&& msk, tli, arr)

let smoothsNR p =
//  if p < 2 then Seq.singleton (bigint 1) else
let smthprms = primesTo p |> Seq.rev |> Seq.map bigint
let rstps = Seq.tail smthprms
let frstcis =
let rec nxt n =
CIS(n, fun () -> nxt (n * frstp)) in nxt frstp
let nxt dq =
Seq.initInfinite ((+) 1I << bigint)
let newcis cis p =
let rec nxt (CIS(hd, tlf) as cs) dq =
let nxtq = peekDQ dq
if hd < nxtq then CIS(hd, fun () -> nxt (tlf()) (pushDQ (hd * p) dq))
else CIS(nxtq, fun () -> nxt cs (pushDQ (nxtq * p) dq |> pullDQ))
CIS(p, fun () -> nxt cis (makeDQ (p * p)))
CIS(1I, fun () -> Seq.fold newcis frstcis rstps)

let strt = System.DateTime.Now.Ticks

primesTo 29 |> Seq.iter (fun p ->
printfn "First 25 %d-smooth:" p
smoothsNR p |> stringCIS 25 |> printfn "%s\r\n")

primesTo 29 |> Seq.skip 1 |> Seq.iter (fun p ->
printfn "The first three from the 3,000th %d-smooth numbers are:" p
smoothsNR p |> skipCIS 2999 |> stringCIS 3 |> printfn "%s\r\n")

primesTo 521 |> Seq.skipWhile ((>) 503) |> Seq.iter (fun p ->
printfn "The first 20 from the 30,000th up %d-smooth numbers are:" p
smoothsNR p |> skipCIS 29999 |> stringCIS 20 |> printfn "%s\r\n")

let stop = System.DateTime.Now.Ticks
printfn "This took %d milliseconds." ((stop - strt) / 10000L)
```

## Factor

```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

```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
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 = 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]
```

The solution is based on the hamming numbers solution Hamming_numbers#Avoiding_generation_of_duplicates.

```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

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

```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) ) {
System.out.printf("The first %d %d-smooth numbers:%n", max, n);
BigInteger[] humble = nSmooth(max, primes.toArray(new BigInteger));
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<>();
for ( int n = 3 ; n <= 29 ; n++ ) {
if ( isPrime(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));
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) ) {
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));
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 = 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;
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

```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[p]

for (let j = 1; j < p.length; j++){
if (p[j][p[j]] < m) m = p[j][p[j]]
}

for (let j = 0; j < p.length; j++){
p[j].push(p[j]*m)
if (p[j][p[j]] == m) p[j]++
}

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; n <= x; n++){
if (isPrime(n)){
sOut += x + " to " + (x - 1 + x) + " " + n + "-smooth numbers: " + smooth(n, x, x) + "\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
```

## jq

Works with gojq, the Go implementation of jq

The output shown below is from a run of gojq, which has unbounded-precision arithmetic.

The C implementation of jq produces the same results for the first task (namely showing the first 125 n-smooth numbers for n <= 29), but not for the subsequent task.

```def select_while(s; cond):
label \$out | s | if cond then . else break \$out end;

### Some primes
# 168 small primes
def small_primes:  [
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,
101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199,
211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293,
307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397,
401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499,
503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599,
601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691,
701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797,
809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887,
907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997
];

### n-smooth numbers
def nSmooth(\$n; \$size):
small_primes[-1] as \$maxn
| if \$n < 2 or \$n > \$maxn then "nSmooth: n must be in 2 .. \(\$maxn) inclusive" | error
elif (\$size < 1) then "nSmooth: size must be at least 1" | error
elif any(small_primes[]; \$n == .) | not then "nSmooth: n must be a prime number" | error
else (small_primes|length) as \$length
| {ns: [1, (range(1; \$size)|null)],
i: 0,
next: [] }
| until(.done or .i == \$length;
if (small_primes[.i] > \$n) then .done = true
else .next += [small_primes[.i]]
| .i += 1
end )
| .indices = [range(0; .next|length) | 0]
| reduce range(1; \$size) as \$m (.;
.ns[\$m] = (.next | min)
| reduce range(0; .indices|length) as \$i (.;
if (.ns[\$m] == .next[\$i])
then .indices[\$i] += 1
| .next[\$i] = small_primes[\$i] * .ns[.indices[\$i]]
else .
end ))
| .ns
end;

[select_while(small_primes[]; . <= 29)] as \$smallPrimes
| (\$smallPrimes[]
| "\nThe first 25 \(.)-smooth numbers are:",
nSmooth(.; 25)),
"",
(\$smallPrimes[1:][]
| "\nThe 3,000th to 3,202nd \(.)-smooth numbers are:",
nSmooth(.; 3002)[2999:] ),
"",
( (503, 509, 521)
|"\nThe 30,000th to 30,019th \(.)-smooth numbers are:",
nSmooth(.; 30019)[29999:] ) ;

Output:

Invocation: gojq -nrcf n-smooth-numbers.jq

```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]
```

## Julia

```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

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)
var i = three
while (i <= p521) {
if (i.isProbablePrime(1)) {
if (i <= p29) {
}
}
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 = BigInteger.ONE
val next = mutableListOf<BigInteger>()
for (i in 0 until primes.size) {
if (primes[i] > bn) {
break
}
}
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

```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];
s[[30000 ;; 30019]]
s = Select[Range[10^5], FactorInteger /* Last /* First /* LessEqualThan];
s[[30000 ;; 30019]]
s = Select[Range[10^5], FactorInteger /* Last /* First /* LessEqualThan];
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

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:
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 = newInt(1)

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

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
```

## PARI/GP

```Vi__smooth(V0, C)= {
my( W= #V0, V_r= Vec(,C), v= V0, ix= vector(W,i,1), t);
for( c= 2, C
, V_r[c]= t= vecmin(v);
for( w= 1, W
, if( v[w] == t, v[w]= V0[w] * V_r[ ix[w]++ ]);
);
);
V_r;
}
N_smooth(N, C=20, S=1)= Vi__smooth(primes([0,N]),S+C-1)[S..S+C-1];

[print(v) |v<-[N_smooth(N, 25)        |N<-primes([  2, 29])]];
[print(v) |v<-[N_smooth(N,  3,  3000) |N<-primes([  3, 29])]];
[print(v) |v<-[N_smooth(N, 20, 30000) |N<-primes([503,521])]];```
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]
```

## Pascal

### 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 - ;

{ 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```

### Free Pascal

misunderstood limit 3000 as n-smooth number index. So 2-smooth (3000) = 2^n was out of reach.

```program HammNumb;
{\$IFDEF FPC} {\$MODE DELPHI} {\$OPTIMIZATION ON,ALL}{\$ENDIF}
{\$IFDEF WINDOWS}{\$APPTYPE CONSOLE}{\$ENDIF}
type
tHamNum = record
hampot : array[0..167] of Word;
hampotmax,
hamNum : NativeUint;
end;
const
primes : array[0..167] of word =
(2, 3, 5, 7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71
,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151
,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233
,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317
,331,337,347,349,353,359,367,373,379,383,389,397,401,409,419
,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503
,509,521,523,541,547,557,563,569,571,577,587,593,599,601,607
,613,617,619,631,641,643,647,653,659,661,673,677,683,691,701
,709,719,727,733,739,743,751,757,761,769,773,787,797,809,811
,821,823,827,829,839,853,857,859,863,877,881,883,887,907,911
,919,929,937,941,947,953,967,971,977,983,991,997);
var
HNum:tHamNum;

procedure OutHamNum(const HNum:tHamNum);
var
i : NativeInt;
Begin
with Hnum do
Begin
write(hamNum:12,' : ');
For i := 0 to hampotmax-1 do
Begin
if hampot[i] >0 then
if hampot[i] = 1 then
write(primes[i],'*')
else
write(primes[i],'^',hampot[i],'*');
end;
if hampot[hampotmax] >0 then
begin
write(primes[hampotmax]);
if hampot[hampotmax] > 1 then
write('^',hampot[hampotmax]);
end;
end;
writeln;
end;

procedure NextHammNum(var HNum:tHamNum;maxP:NativeInt);
var
q,p,nr,n,pIdx,momPrime : NativeUInt;
begin
//special case prime = 2
IF maxP = 0 then
begin
IF HNum.hampot <> 0 then
HNum.hamNum *= 2
else
HNum.hamNum := 1;
inc(HNum.hampot);
EXIT;
end;

n := HNum.hamNum;
repeat
inc(n);
nr := n;
pIdx := 0;
repeat
momPrime := primes[pIdx];
q := nr div momPrime;
p := 0;
While q*momPrime=nr do
Begin
inc(p);
nr := q;
q := nr div momPrime;
end;
HNum.hampot[pIdx] := p;
inc(pIdx);
until (nr=1) OR (pIdx > maxp)
//found one, than finished
until nr = 1;

With HNum do
Begin
hamNum := n;
hamPotmax := pIdx-1;
end;
end;

procedure OutXafterYSmooth(X,Y,SmoothIdx: NativeUInt);
var
i: NativeUint;
begin
IF SmoothIdx> High(primes) then
EXIT;
fillChar(HNum,SizeOf(HNum),#0);

i := 0;
While HNum.HamNum < Y do
NextHammNum(HNum,SmoothIdx);

write('first ',X,' after ',Y,' ',primes[SmoothIdx]:3,'-smooth numbers : ');
IF x >10 then
writeln;

for i := 1 to X-1 do
begin
write(HNum.HamNum,' ');
NextHammNum(HNum,SmoothIdx);
end;
writeln(HNum.HamNum,' ');
end;

var
j: NativeUint;
Begin
j := 0;
while primes[j] <= 29 do
Begin
OutXafterYSmooth(25,1,j);
inc(j);
end;
writeln;

j := 1;
while primes[j] <= 29 do
Begin
OutXafterYSmooth(3,3000,j);
inc(j);
end;
writeln;

while primes[j] < 503 do
inc(j);
while primes[j] <= 521 do
Begin
OutXafterYSmooth(20,30000,j);
OutHamNum(Hnum);
inc(j);
end;
writeln;
End.
```
Output:
```first 25 after 1   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 after 1   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 after 1   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 after 1   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 after 1  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 after 1  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 after 1  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 after 1  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 after 1  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 after 1  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

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

first 20 after 30000 503-smooth numbers :
30000 30003 30005 30008 30012 30014 30015 30016 30020 30024 30030 30033 30039 30044 30046 30048 30049 30051 30056 30057
30057 : 3*43*233
first 20 after 30000 509-smooth numbers :
30000 30003 30005 30008 30012 30014 30015 30016 30020 30024 30030 30031 30033 30039 30044 30046 30048 30049 30051 30056
30056 : 2^3*13*17^2
first 20 after 30000 521-smooth numbers :
30000 30003 30005 30008 30012 30014 30015 30016 30020 30024 30030 30031 30033 30039 30044 30046 30048 30049 30051 30056
30056 : 2^3*13*17^2```

## Perl

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,  for 0..\$#m;

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

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

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 =  * size
ns = 1

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

indicies =  * 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

```  [ 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

(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

```/*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

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 =  * limit
ns = 1
nextp = primes[0..primes.index(n)]

indices =  * 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

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

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

```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 = 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

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 = BigInt.one
var next = []
for (i in 0...primes.count) {
if (primes[i] > bn) break
}
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

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
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