# Jordan-Pólya numbers

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

Jordan-Pólya numbers (or J-P numbers for short) are the numbers that can be obtained by multiplying together one or more (not necessarily distinct) factorials.

Example

480 is a J-P number because 480 = 2! x 2! x 5!.

What is the largest J-P number less than 100 million?

Bonus

Find and show on this page the 800th, 1,800th, 2,800th and 3,800th J-P numbers and also show their decomposition into factorials in highest to lowest order. Optionally, do the same for the 1,050th J-P number.

Where there is more than one way to decompose a J-P number into factorials, choose the way which uses the largest factorials.

Hint: These J-P numbers are all less than 2^53.

References

## C

Translation of: Wren
Library: GLib

A translation of the second version. Run time about 0.035 seconds.

```#include <stdio.h>
#include <stdint.h>
#include <stdbool.h>
#include <locale.h>
#include <glib.h>

uint64_t factorials[19] = {1, 1};

void cacheFactorials() {
uint64_t fact = 1;
int i;
for (i = 2; i < 19; ++i) {
fact *= i;
factorials[i] = fact;
}
}

int findNearestFact(uint64_t n) {
int i;
for (i = 1; i < 19; ++i) {
if (factorials[i] >= n) return i;
}
return 18;
}

int findNearestInArray(GArray *a, uint64_t n) {
int l = 0, r = a->len, m;
while (l < r) {
m = (l + r)/2;
if (g_array_index(a, uint64_t, m) > n) {
r = m;
} else {
l = m + 1;
}
}
if (r > 0 && g_array_index(a, uint64_t, r-1) == n) return r - 1;
return r;
}

GArray *jordanPolya(uint64_t limit) {
int i, ix, k, l, p;
uint64_t t, rk, kl;
GArray *res = g_array_new(false, false, sizeof(uint64_t));
ix = findNearestFact(limit);
for (i = 0; i <= ix; ++i) {
t = factorials[i];
g_array_append_val(res, t);
}
k = 2;
while (k < res->len) {
rk = g_array_index(res, uint64_t, k);
for (l = 2; l < res->len; ++l) {
t = g_array_index(res, uint64_t, l);
if (t > limit/rk) break;
kl = t * rk;
while (true) {
p = findNearestInArray(res, kl);
if (p < res->len && g_array_index(res, uint64_t, p) != kl) {
g_array_insert_val(res, p, kl);
} else if (p == res->len) {
g_array_append_val(res, kl);
}
if (kl > limit/rk) break;
kl *= rk;
}
}
++k;
}
return g_array_remove_index(res, 0);
}

GArray *decompose(uint64_t n, int start) {
uint64_t i, s, t, m, prod;
GArray *f, *g;
for (s = start; s > 0; --s) {
f = g_array_new(false, false, sizeof(uint64_t));
if (s < 2) return f;
m = n;
while (!(m % factorials[s])) {
g_array_append_val(f, s);
m /= factorials[s];
if (m == 1) return f;
}
if (f->len > 0) {
g = decompose(m, s - 1);
if (g->len > 0) {
prod = 1;
for (i = 0; i < g->len; ++i) {
prod *= factorials[(int)g_array_index(g, uint64_t, i)];
}
if (prod == m) {
for (i = 0; i < g->len; ++i) {
t = g_array_index(g, uint64_t, i);
g_array_append_val(f, t);
}
g_array_free(g, true);
return f;
}
}
g_array_free(g, true);
}
g_array_free(f, true);
}
}

char *superscript(int n) {
char* ss[] = {"⁰", "¹", "²", "³", "⁴", "⁵", "⁶", "⁷", "⁸", "⁹"};
if (n < 10) return ss[n];
static char buf[7];
sprintf(buf, "%s%s", ss[n/10], ss[n%10]);
return buf;
}

int main() {
int i, j, ix, count;
uint64_t t, u;
GArray *v, *w;
cacheFactorials();
v = jordanPolya(1ull << 53);
printf("First 50 Jordan-Pólya numbers:\n");
for (i = 0; i < 50; ++i) {
printf("%4ju ", g_array_index(v, uint64_t, i));
if (!((i + 1) % 10)) printf("\n");
}
printf("\nThe largest Jordan-Pólya number before 100 millon: ");
setlocale(LC_NUMERIC, "");
ix = findNearestInArray(v, 100000000ull);
printf("%'ju\n\n", g_array_index(v, uint64_t, ix - 1));

uint64_t targets[5] = {800, 1050, 1800, 2800, 3800};
for (i = 0; i < 5; ++i) {
t = g_array_index(v, uint64_t, targets[i] - 1);
printf("The %'juth Jordan-Pólya number is : %'ju\n", targets[i], t);
w = decompose(t, 18);
count = 1;
t = g_array_index(w, uint64_t, 0);
printf(" = ");
for (j = 1; j < w->len; ++j) {
u = g_array_index(w, uint64_t, j);
if (u != t) {
if (count == 1) {
printf("%ju! x ", t);
} else {
printf("(%ju!)%s x ", t, superscript(count));
count = 1;
}
t = u;
} else {
++count;
}
}
if (count == 1) {
printf("%ju! x ", t);
} else {
printf("(%ju!)%s x ", t, superscript(count));
}
printf("\b\b \n\n");
g_array_free(w, true);
}
g_array_free(v, true);
return 0;
}
```
Output:
```First 50 Jordan-Pólya numbers:
1    2    4    6    8   12   16   24   32   36
48   64   72   96  120  128  144  192  216  240
256  288  384  432  480  512  576  720  768  864
960 1024 1152 1296 1440 1536 1728 1920 2048 2304
2592 2880 3072 3456 3840 4096 4320 4608 5040 5184

The largest Jordan-Pólya number before 100 millon: 99,532,800

The 800th Jordan-Pólya number is : 18,345,885,696
= (4!)⁷ x (2!)²

The 1,050th Jordan-Pólya number is : 139,345,920,000
= 8! x (5!)³ x 2!

The 1,800th Jordan-Pólya number is : 9,784,472,371,200
= (6!)² x (4!)² x (2!)¹⁵

The 2,800th Jordan-Pólya number is : 439,378,587,648,000
= 14! x 7!

The 3,800th Jordan-Pólya number is : 7,213,895,789,838,336
= (4!)⁸ x (2!)¹⁶
```

## C++

```#include <algorithm>
#include <cmath>
#include <cstdint>
#include <iomanip>
#include <iostream>
#include <map>
#include <set>
#include <unordered_map>
#include <vector>

constexpr int64_t LIMIT = static_cast<uint64_t>(1) << 53;

std::set<int64_t> jordan_polya_set;
std::unordered_map<int64_t, std::map<int32_t, int32_t>> decompositions;

std::string toString(const std::map<int32_t, int32_t>& a_map) {
std::string result;
for ( const auto& [key, value] : a_map ) {
result = std::to_string(key) + "!" + ( value == 1 ? "" : "^" + std::to_string(value) ) + " * " + result;
}
return result.substr(0, result.length() - 3);
}

std::vector<int64_t> set_to_vector(const std::set<int64_t>& a_set) {
std::vector<int64_t> result;
result.reserve(a_set.size());

for ( const int64_t& element : a_set ) {
result.emplace_back(element);
}
return result;
}

void insert_or_update(std::map<int32_t, int32_t>& map, const int32_t& entry) {
if ( map.find(entry) == map.end() ) {
map.emplace(entry, 1);
} else {
map[entry]++;
}
}

void create_jordan_polya() {
jordan_polya_set.emplace(1);
decompositions[1] = std::map<int32_t, int32_t>();
int64_t factorial = 1;

for ( int32_t multiplier = 2; multiplier <= 20; ++multiplier ) {
factorial *= multiplier;
for ( int64_t number : jordan_polya_set ) {
while ( number <= LIMIT / factorial ) {
int64_t original = number;
number *= factorial;
jordan_polya_set.emplace(number);
decompositions[number] = decompositions[original];
insert_or_update(decompositions[number], multiplier);
}
}
}
}

int main() {
create_jordan_polya();

std::vector<int64_t> jordan_polya = set_to_vector(jordan_polya_set);

std::cout << "The first 50 Jordan-Polya numbers:" << std::endl;
for ( int64_t i = 0; i < 50; ++i ) {
std::cout << std::setw(5) << jordan_polya[i] << ( i % 10 == 9 ? "\n" : "" );
}

const std::vector<int64_t>::iterator hundred_million =
std::lower_bound(jordan_polya.begin(), jordan_polya.end(), 100'000'000);
std::cout << "\n" << "The largest Jordan-Polya number less than 100 million: "
<< jordan_polya[(hundred_million - jordan_polya.begin() - 1)] << std::endl << std::endl;

for ( int32_t i : { 800, 1050, 1800, 2800, 3800 } ) {
std::cout << "The " << i << "th Jordan-Polya number is: " << jordan_polya[i - 1]
<< " = " << toString(decompositions[jordan_polya[i - 1]]) << std::endl;
}
}
```
Output:
```The first 50 Jordan-Polya numbers:
1    2    4    6    8   12   16   24   32   36
48   64   72   96  120  128  144  192  216  240
256  288  384  432  480  512  576  720  768  864
960 1024 1152 1296 1440 1536 1728 1920 2048 2304
2592 2880 3072 3456 3840 4096 4320 4608 5040 5184

The largest Jordan-Polya number less than 100 million: 99532800

The 800th Jordan-Polya number is: 18345885696 = 4!^7 * 2!^2
The 1050th Jordan-Polya number is: 139345920000 = 8! * 5!^3 * 2!
The 1800th Jordan-Polya number is: 9784472371200 = 6!^2 * 4!^2 * 2!^15
The 2800th Jordan-Polya number is: 439378587648000 = 14! * 7!
The 3800th Jordan-Polya number is: 7213895789838336 = 4!^8 * 2!^16
```

## EasyLang

Translation of: FreeBASIC
```fastfunc jpnum m .
n = m
limite = 7
while 1 = 1
fac = 1
i = 1
while i < limite
i += 1
fac *= i
.
repeat
q = n div fac
if n mod fac = 0
if q = 1
return 1
.
n = q
else
fac = fac / i
i -= 1
.
until i = 1
.
limite -= 1
if limite = 0
return 0
.
n = m
.
.
numfmt 0 5
write 1
c = 1
n = 2
repeat
if jpnum n = 1
c += 1
if c <= 50
write n
if c mod 10 = 0
print ""
.
.
sn = n
.
n += 2
until n >= 1e8
.
print ""
print "The largest Jordan-Polya number before 100 million: " & sn```

## FreeBASIC

Translation of: XPL0

Simple-minded brute force. No bonus.

```Dim Shared As Uinteger Factorials(1+12)

Function isJPNum(m As Uinteger) As Boolean
Dim As Uinteger n = m, limite = 7, i, q
Do
i = limite
Do
q = n / Factorials(i)
If n Mod Factorials(i) = 0 Then
If q = 1 Then Return True
n = q
Else
i -= 1
End If
If i = 1 Then
If limite = 1 Then Return False
limite -= 1
n = m
Exit Do
End If
Loop
Loop
End Function

Dim As Uinteger fact = 1, n
For n = 1 To 12
fact *= n
Factorials(n) = fact
Next

Print "First 50 Jordan-Polya numbers:"
Print "    1";
Dim As Uinteger c, sn
c = 1
n = 2
Do
If isJPNum(n) Then
c += 1
If c <= 50 Then
Print Using "#####"; n;
If c Mod 10 = 0 Then Print
End If
sn = n
End If
n += 2
Loop Until n >= 1e8

Print !"\nThe largest Jordan-Polya number before 100 million: "; sn

Sleep```
Output:
`Same as XPL0 entry.`

## Go

Translation of: C
Library: Go-rcu

```package main

import (
"fmt"
"rcu"
"sort"
)

var factorials = make([]uint64, 19)

func cacheFactorials() {
factorials[0] = 1
for i := uint64(1); i < 19; i++ {
factorials[i] = factorials[i-1] * i
}
}

func jordan_polya(limit uint64) []uint64 {
ix := sort.Search(19, func(i int) bool { return factorials[i] >= limit })
if ix > 18 {
ix = 18
}
var res []uint64
res = append(res, factorials[0:ix+1]...)
k := 2
for k < len(res) {
rk := res[k]
for l := 2; l < len(res); l++ {
t := res[l]
if t > limit/rk {
break
}
kl := t * rk
for {
p := sort.Search(len(res), func(i int) bool { return res[i] >= kl })
if p < len(res) && res[p] != kl {
res = append(res[0:p+1], res[p:]...)
res[p] = kl
} else if p == len(res) {
res = append(res, kl)
}
if kl > limit/rk {
break
}
kl *= rk
}
}
k++
}
return res[1:]
}

func decompose(n uint64, start int) []uint64 {
for s := uint64(start); s > 0; s-- {
var f []uint64
if s < 2 {
return f
}
m := n
for m%factorials[s] == 0 {
f = append(f, s)
m /= factorials[s]
if m == 1 {
return f
}
}
if len(f) > 0 {
g := decompose(m, int(s-1))
if len(g) > 0 {
prod := uint64(1)
for _, e := range g {
prod *= factorials[e]
}
if prod == m {
return append(f, g...)
}
}
}
}
return []uint64{}
}

func superscript(n int) string {
ss := []string{"⁰", "¹", "²", "³", "⁴", "⁵", "⁶", "⁷", "⁸", "⁹"}
if n < 10 {
return ss[n]
}
return ss[n/10] + ss[n%10]
}

func main() {
cacheFactorials()
v := jordan_polya(uint64(1) << 53)
fmt.Println("First 50 Jordan-Pólya numbers:")
for i := 0; i < 50; i++ {
fmt.Printf("%4d ", v[i])
if (i+1)%10 == 0 {
fmt.Println()
}
}
fmt.Printf("\nThe largest Jordan-Pólya number before 100 million: ")
ix := sort.Search(len(v), func(i int) bool { return v[i] >= 100_000_000 })
fmt.Println(rcu.Commatize(v[ix-1]))
fmt.Println()
for _, e := range []uint64{800, 1050, 1800, 2800, 3800} {
fmt.Printf("The %sth Jordan-Pólya number is : %s\n", rcu.Commatize(e), rcu.Commatize(v[e-1]))
w := decompose(v[e-1], 18)
count := 1
t := w[0]
fmt.Printf(" = ")
for j := 1; j < len(w); j++ {
u := w[j]
if u != t {
if count == 1 {
fmt.Printf("%d! x ", t)
} else {
fmt.Printf("(%d!)%s x ", t, superscript(count))
count = 1
}
t = u
} else {
count++
}
}
if count == 1 {
fmt.Printf("%d! x ", t)
} else {
fmt.Printf("(%d!)%s x ", t, superscript(count))
}
fmt.Printf("\b\b \n\n")
}
}
```
Output:
```Same as C example.
```

## J

```F=. !P=. p:i.100x
jpprm=: P{.~F I. 1+]

Fs=. 2}.!i.1+{:P
jpfct=: Fs |.@:{.~ Fs I. 1+]

isjp=: {{
if. 2>y do. y return.
elseif. 0 < #(q:y)-.jpprm y do. 0 return.
else.
for_f. (#~ ] = <.) (%jpfct) y do.
if. isjp f do. 1 return. end.
end.
end.
0
}}"0

showjp=: {{
if. 2>y do. i.0 return. end.
F=. f{~1 i.~b #inv isjp Y#~b=. (]=<.) Y=. y%f=. jpfct y
F,showjp y%F
}}

NB. generate a Jordan-Pólya of the given length
jpseq=: {{
r=. 1 2x   NB. sequence, so far
f=. 2 6x   NB. factorial factors
i=. 1 0    NB. index of next item of f for each element of r
g=. 6 4x   NB. product of r with selected item of f
while. y>#r do.
r=. r, nxt=. <./g  NB. next item in r
j=. I.b=. g=nxt    NB. items of g which just be recalculated
if. nxt={:f do.    NB. need new factorial factor/
f=. f,!2+#f
end.
i=. 0,~i+b         NB. update indices into f
g=. (2*nxt),~((j{r)*((<:#f)<.j{i){f) j} g
end.
y{.r
}}
```

```   5 10\$jpseq 50
1    2    4    6    8   12   16   24   32   36
48   64   72   96  120  128  144  192  216  240
256  288  384  432  480  512  576  720  768  864
960 1024 1152 1296 1440 1536 1728 1920 2048 2304
2592 2880 3072 3456 3840 4096 4320 4608 5040 5184
<:^:(0=isjp)^:_]1e8
99532800
showjp 99532800
720 720 24 2 2 2
```

Note that jp factorizations are not necessarily unique. For example, 120 120 6 6 6 2 2 2 2 2 would also be a jp factorization of 99532800.

Bonus (indicated numbers from jp sequence, followed by a jp factorization):

```   s=: jpseq 4000
(,showjp) (<:800){s
18345885696 24 24 24 24 24 24 24 2 2
(,showjp) (<:1800){s
9784472371200 720 720 24 24 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
(,showjp) (<:2800){s
439378587648000 87178291200 5040
(,showjp) (<:3800){s
7213895789838336 24 24 24 24 24 24 24 24 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
```

## Java

```import java.util.ArrayList;
import java.util.HashMap;
import java.util.Iterator;
import java.util.List;
import java.util.Map;
import java.util.Set;
import java.util.TreeMap;
import java.util.TreeSet;

public final class JordanPolyaNumbers {

public static void main(String[] aArgs) {
createJordanPolya();

final long belowHundredMillion = jordanPolyaSet.floor(100_000_000L);
List<Long> jordanPolya = new ArrayList<Long>(jordanPolyaSet);

System.out.println("The first 50 Jordan-Polya numbers:");
for ( int i = 0; i < 50; i++ ) {
System.out.print(String.format("%5s%s", jordanPolya.get(i), ( i % 10 == 9 ? "\n" : "" )));
}
System.out.println();

System.out.println("The largest Jordan-Polya number less than 100 million: " + belowHundredMillion);
System.out.println();

for ( int i : List.of( 800, 1050, 1800, 2800, 3800 ) ) {
System.out.println("The " + i + "th Jordan-Polya number is: " + jordanPolya.get(i - 1)
+ " = " + toString(decompositions.get(jordanPolya.get(i - 1))));
}
}

private static void createJordanPolya() {
Set<Long> nextSet = new TreeSet<Long>();
decompositions.put(1L, new TreeMap<Integer, Integer>());
long factorial = 1;

for ( int multiplier = 2; multiplier <= 20; multiplier++ ) {
factorial *= multiplier;
for ( Iterator<Long> iterator = jordanPolyaSet.iterator(); iterator.hasNext(); ) {
long number = iterator.next();
while ( number <= Long.MAX_VALUE / factorial ) {
long original = number;
number *= factorial;
decompositions.put(number, new TreeMap<Integer, Integer>(decompositions.get(original)));
decompositions.get(number).merge(multiplier, 1, Integer::sum);
}
}
nextSet.clear();
}
}

private static String toString(Map<Integer, Integer> aMap) {
String result = "";
for ( int key : aMap.keySet() ) {
result = key + "!" + ( aMap.get(key) == 1 ? "" :"^" + aMap.get(key) ) + " * " + result;
}
return result.substring(0, result.length() - 3);
}

private static TreeSet<Long> jordanPolyaSet = new TreeSet<Long>();
private static Map<Long, Map<Integer, Integer>> decompositions = new HashMap<Long, Map<Integer, Integer>>();

}
```
Output:
```The first 50 Jordan-Polya numbers:
1    2    4    6    8   12   16   24   32   36
48   64   72   96  120  128  144  192  216  240
256  288  384  432  480  512  576  720  768  864
960 1024 1152 1296 1440 1536 1728 1920 2048 2304
2592 2880 3072 3456 3840 4096 4320 4608 5040 5184

The largest Jordan-Polya number less than 100 million: 99532800

The 800th Jordan-Polya number is: 18345885696 = 4!^7 * 2!^2
The 1050th Jordan-Polya number is: 139345920000 = 8! * 5!^3 * 2!
The 1800th Jordan-Polya number is: 9784472371200 = 6!^2 * 4!^2 * 2!^15
The 2800th Jordan-Polya number is: 439378587648000 = 14! * 7!
The 3800th Jordan-Polya number is: 7213895789838336 = 4!^8 * 2!^16
```

## jq

Works with: jq

Also works with gojq, the Go implementation of jq

```### Generic functions
# For gojq
def _nwise(\$n):
def n: if length <= \$n then . else .[0:\$n] , (.[\$n:] | n) end;
n;

def lpad(\$len): tostring | (\$len - length) as \$l | (" " * \$l) + .;

# tabular print
def tprint(columns; wide):
reduce _nwise(columns) as \$row ("";
. + (\$row|map(lpad(wide)) | join(" ")) + "\n" );

# Input: an array
# Output: a stream of pairs [\$x, \$frequency]
# A two-level dictionary is used: .[type][tostring]
def frequencies:
if length == 0 then empty
else . as \$in
| reduce range(0; length) as \$i ({};
\$in[\$i] as \$x
| .[\$x|type][\$x|tostring] as \$pair
| if \$pair
then .[\$x|type][\$x|tostring] |= (.[1] += 1)
else .[\$x|type][\$x|tostring] = [\$x, 1]
end )
| .[][]
end ;

# Output: the items in the stream up to but excluding the first for which cond is truthy
def emit_until(cond; stream): label \$out | stream | if cond then break \$out else . end;

### Jordan-Pólya numbers
# input: {factorial}
# output: an array
def JordanPolya(\$lim; \$mx):
if \$lim < 2 then [1]
else . + {v: [1], t: 1, k: 2}
| .mx = (\$mx // \$lim)
| until(.k > .mx or .t > \$lim;
.t *= .k
| if .t <= \$lim
then reduce JordanPolya((\$lim/.t)|floor; .t)[] as \$rest (.;
.v += [.t * \$rest] )
| .k += 1
else .
end)
| .v
| unique
end;

# Cache m! for m <= \$n
def cacheFactorials(\$n):
{fact: 1, factorial: [1]}
| reduce range(1; \$n + 1) as \$i (.;
.fact *= \$i
| .factorial[\$i] = .fact );

# input: {factorial}
def Decompose(\$n; \$start):
if \$start and \$start < 2 then []
else
{ factorial,
start: (\$start // 18),
m: \$n,
f: [] }
| label \$out
| foreach range(.start; 1; -1) as \$i (.;
.i = \$i
| .emit = null
| until (.emit or (.m % .factorial[\$i] != 0);
.f += [\$i]
| .m = (.m / .factorial[\$i])
| if .m == 1 then .emit = .f else . end)
| if .emit then ., break \$out else . end)
| if .emit then .emit
elif .i == 2 then Decompose(\$n; .start-1)
else empty
end
end;

# Input: {factorial}
# \$v should be an array of J-P numbers
def synopsis(\$v):
(100, 800, 1800, 2800, 3800) as \$i
| if \$v[\$i-1] == null
else "\nThe \(\$i)th Jordan-Pólya number is \(\$v[\$i-1] )",
([Decompose(\$v[\$i-1]; null) | frequencies]
| map( if (.[1] == 1) then "\(.[0])!"  else "(\(.[0])!)^\(.[1])" end)
| "  i.e. " + join(" * ") )
end ;

cacheFactorials(18)
| JordanPolya(pow(2;53)-1; null) as \$v
| "\(\$v|length) Jordan–Pólya numbers have been found. The first 50 are:",
( \$v[:50] | tprint(10; 4)),
"\nThe largest Jordan–Pólya number before 100 million: " +
"\(if \$v[-1] > 1e8 then last(emit_until(. >= 1e8; \$v[])) else "not found" end)",
synopsis(\$v) ;

Output:

gojq and jq produce the same results except that gojq produces the factorizations in a different order. The output shown here corresponds to the invocation: jq -nr -f jordan-polya-numbers.jq

```3887 Jordan–Pólya numbers have been found. The first 50 are:
1    2    4    6    8   12   16   24   32   36
48   64   72   96  120  128  144  192  216  240
256  288  384  432  480  512  576  720  768  864
960 1024 1152 1296 1440 1536 1728 1920 2048 2304
2592 2880 3072 3456 3840 4096 4320 4608 5040 5184

The largest Jordan–Pólya number before 100 million: 99532800

The 100th Jordan-Pólya number is 92160
i.e. 6! * (2!)^7

The 800th Jordan-Pólya number is 18345885696
i.e. (4!)^7 * (2!)^2

The 1800th Jordan-Pólya number is 9784472371200
i.e. (6!)^2 * (4!)^2 * (2!)^15

The 2800th Jordan-Pólya number is 439378587648000
i.e. 14! * 7!

The 3800th Jordan-Pólya number is 7213895789838336
i.e. (4!)^8 * (2!)^16
```

## Julia

```function aupto(limit::T) where T <: Integer
res = map(factorial, T(1):T(18))
k = 2
while k < length(res)
rk = res[k]
for j = 2:length(res)
kl = res[j] * rk
kl > limit && break
while kl <= limit && kl ∉ res
push!(res, kl)
kl *= rk
end
end
k += 1
end
return sort!((sizeof(T) > sizeof(Int) ? T : Int).(res))[begin+1:end]
end

const factorials = map(factorial, 2:18)

""" Factor a J-P number into a smallest vector of factorials and their powers """
function factor_as_factorials(n::T) where T <: Integer
fac_exp = Tuple{Int, Int}[]
for idx in length(factorials):-1:1
m = n
empty!(fac_exp)
for i in idx:-1:1
expo = 0
while m % factorials[i] == 0
expo += 1
m ÷= factorials[i]
end
if expo > 0
push!(fac_exp, (i + 1, expo))
end
end
m == 1 && break
end
return fac_exp
end

const superchars = ["\u2070", "\u00b9", "\u00b2", "\u00b3", "\u2074",
"\u2075", "\u2076", "\u2077", "\u2078", "\u2079"]
""" Express a positive integer as Unicode superscript digit characters """
super(n::Integer) = prod(superchars[i + 1] for i in reverse(digits(n)))

arr = aupto(2^53)

println("First 50 Jordan–Pólya numbers:")
foreach(p -> print(rpad(p[2], 6), p[1] % 10 == 0 ? "\n" : ""), enumerate(arr[1:50]))

println("\nThe largest Jordan–Pólya number before 100 million: ", arr[findlast(<(100_000_000), arr)])

for n in [800, 1800, 2800, 3800]
print("\nThe \$(n)th Jordan-Pólya number is: \$(arr[n])\n= ")
println(join(map(t -> "\$(t[1])!\$(t[2] > 1 ? super(t[2]) : "")",
factor_as_factorials(arr[n])), " x "))
end
```
Output:
```First 50 Jordan–Pólya numbers:
1     2     4     6     8     12    16    24    32    36
48    64    72    96    120   128   144   192   216   240
256   288   384   432   480   512   576   720   768   864
960   1024  1152  1296  1440  1536  1728  1920  2048  2304
2592  2880  3072  3456  3840  4096  4320  4608  5040  5184

The largest Jordan–Pólya number before 100 million: 99532800

The 800th Jordan-Pólya number is: 18345885696
= 4!⁷ x 2!²

The 1800th Jordan-Pólya number is: 9784472371200
= 6!² x 4!² x 2!¹⁵

The 2800th Jordan-Pólya number is: 439378587648000
= 14! x 7!

The 3800th Jordan-Pólya number is: 7213895789838336
= 4!⁸ x 2!¹⁶
```

## Nim

```import std/[algorithm, math, sequtils, strformat, strutils, tables]

const Max = if sizeof(int) == 8: 20 else: 12

type Decomposition = CountTable[int]

const Superscripts: array['0'..'9', string] = ["⁰", "¹", "²", "³", "⁴", "⁵", "⁶", "⁷", "⁸", "⁹"]

func superscript(n: Natural): string =
## Return the Unicode string to use to represent an exponent.
if n == 1:
return ""
for d in \$n:

proc `\$`(d: Decomposition): string =
## Return the representation of a decomposition.
for (value, count) in sorted(d.pairs.toSeq, Descending):

# List of Jordan-Pólya numbers and their decomposition.
var jordanPolya = @[1]
var decomposition: Table[int, CountTable[int]] = {1: initCountTable[int]()}.toTable

# Build the list and the decompositions.
for m in 2..Max:                  # Loop on each factorial.
let f = fac(m)
for k in 0..jordanPolya.high:   # Loop on existing elements.
var n = jordanPolya[k]
while n <= int.high div f:    # Multiply by successive powers of n!
let p = n
n *= f
decomposition[n] = decomposition[p]
decomposition[n].inc(m)

# Sort the numbers and remove duplicates.
jordanPolya = sorted(jordanPolya).deduplicate(true)

echo "First 50 Jordan-Pólya numbers:"
for i in 0..<50:
stdout.write &"{jordanPolya[i]:>4}"
stdout.write if i mod 10 == 9: '\n' else: ' '

echo "\nLargest Jordan-Pólya number less than 100 million: ",
insertSep(\$jordanPolya[jordanPolya.upperBound(100_000_000) - 1])

for i in [800, 1800, 2800, 3800]:
let n = jordanPolya[i - 1]
var d = decomposition[n]
echo &"\nThe {i}th Jordan-Pólya number is:"
echo &"{insertSep(\$n)} = {d}"
```
Output:
```First 50 Jordan-Pólya numbers:
1    2    4    6    8   12   16   24   32   36
48   64   72   96  120  128  144  192  216  240
256  288  384  432  480  512  576  720  768  864
960 1024 1152 1296 1440 1536 1728 1920 2048 2304
2592 2880 3072 3456 3840 4096 4320 4608 5040 5184

Largest Jordan-Pólya number less than 100 million: 99_532_800

The 800th Jordan-Pólya number is:
18_345_885_696 = (4!)⁷(2!)²

The 1800th Jordan-Pólya number is:
9_784_472_371_200 = (6!)²(4!)²(2!)¹⁵

The 2800th Jordan-Pólya number is:
439_378_587_648_000 = (14!)(7!)

The 3800th Jordan-Pólya number is:
7_213_895_789_838_336 = (4!)⁸(2!)¹⁶
```

## Pascal

### Free Pascal

succesive add of next factorial in its power.keep sorted and without doublettes.
Now using Uint64 and only marking which factorial is used, which is unnecessary, but why not. It makes output easier
Runtime for TIO.RUN 127 ms (too short to be significant ) @home 2 ms
Using alternative dblLimit := 1 shl 6 as starting Limit and increase it by a factor of 256
This gets if MaxIdx = 3800 in 4ms.

```program Jordan_Polya_Num;
{\$IFDEF FPC}{\$MODE DELPHI}{\$OPTIMIZATION ON,ALL}{\$COPERATORS ON}{\$ENDIF}
{\$IFDEF Windows}{\$APPTYPE CONSOLE}{\$ENDIF}
uses
sysutils;
const
MaxIdx = 3800;//7279 < 2^62
maxFac = 25;//21!> 2^63
type
tnum = Uint64;
tpow= set of 0..31;// 1==2!^? ,2=3!^? 3=2!^?*3!^?
tFac_mul = packed record
fm_num : tnum;
fm_pow : tpow;
fm_high_idx : word;
fm_high_pow : word;
end;
tpFac_mul = ^tFac_mul;
tFacMulPow = array of tFac_mul;
tFactorial = array[0..maxFac-2] of tnum;

var
FacMulPowGes : tFacMulPow;
Factorial: tFactorial;
LastSearchFor :tFac_mul;
dblLimit : tnum;

function CommatizeUint64(num:Uint64):AnsiString;
var
fromIdx,toIdx :Int32;
Begin
str(num,result);
fromIdx := length(result);
toIdx := fromIdx-1;
if toIdx < 3 then
exit;

toIdx := 4*(toIdx DIV 3)+toIdx MOD 3 +1 ;
setlength(result,toIdx);
repeat
result[toIdx]   := result[FromIdx];
result[toIdx-1] := result[FromIdx-1];
result[toIdx-2] := result[FromIdx-2];
result[toIdx-3] := ',';
dec(toIdx,4);
dec(FromIdx,3);
until FromIdx<=3;
end;

procedure Out_MulFac(idx:Uint32;const fm:tFac_mul);
var
fac,
num : tNum;
FacIdx,pow : integer;
begin
num := fm.fm_num;
FacIdx := fm.fm_high_idx;
write(CommatizeUint64(num):25,' = ');

repeat
pow := 0;
fac := Factorial[FacIdx];
while (num>=fac) AND (num mod Fac = 0) do
Begin
num := num DIV Fac;
inc(pow);
end;
if pow = 0 then
write(' 1')
else
if pow = 1 then
write(' ',FacIdx+2,'!')
else
write(' (',FacIdx+2,'!)^',pow);
if num = 1 then
BREAK;
repeat
dec(FacIdx);
until(FacIdx<0) OR (FacIdx in fm.fm_pow);
until FacIdx < 0;
writeln;

end;

procedure Out_I_th(i: integer);
begin
if i < 0 then
write(i:8,' too small');
if i <= High(FacMulPowGes) then
begin
write(i:6,'-th : ');
Out_MulFac(i,FacMulPowGes[i-1])
end
else
writeln('Too big');
end;

procedure Out_First_N(n: integer);
var
s,fmt : AnsiString;
i,tmp : integer;
Begin
if n<1 then
EXIT;
writeln('The first ',n,' Jordan-Polia numbers');
s := '';
If n > Length(FacMulPowGes) then
n := Length(FacMulPowGes);
dec(n);
tmp := length(CommatizeUint64(FacMulPowGes[n].fm_num))+1;
fmt := '%'+IntToStr(tmp)+'s';
tmp := 72 DIV tmp;
For i := 0 to n do
Begin
s += Format(fmt,[CommatizeUint64(FacMulPowGes[i].fm_num)]);
if (i+1) mod tmp = 0 then
Begin
writeln(s);
s := '';
end;
end;
if s <>'' then
writeln(s);
writeln;
end;

procedure Initfirst;
var
fac: tnum;
i,j,idx: integer;
Begin
fac:= 1;
j := 1;
idx := 0;
For i := 2 to maxFac do
Begin
repeat
inc(j);
fac *= j;
until j = i;
Factorial[idx] := fac;
inc(idx);
end;
Fillchar(LastSearchFor,SizeOf(LastSearchFor),#0);
LastSearchFor.FM_NUM := 0;
//  dblLimit := 1 shl 53;
dblLimit := 1 shl 5;
end;

procedure ResetSearch;
Begin
setlength(FacMulPowGes,0);
end;

procedure GenerateFirst(idx:NativeInt;var res:tFacMulPow);
//generating the first entry with (2!)^n
var
Fac_mul :tFac_mul;
facPow,Fac : tnum;
i,MaxPowOfFac : integer;
begin
fac := Factorial[idx];
MaxPowOfFac := trunc(ln(dblLimit)/ln(Fac))+1;
setlength(res,MaxPowOfFac);

with Fac_Mul do
begin
fm_num := 1;
fm_pow := [0];
fm_high_idx := 0;
end;

res[0] := Fac_Mul;
facPow := 1;
i := 1;
repeat
facPow *= Fac;
if facPow >dblLimit then
BREAK;
with Fac_Mul do
begin
fm_num := facPow;
fm_high_pow := i;
end;
res[i] := Fac_Mul;
inc(i);
until i = MaxPowOfFac;
setlength(res,i);
end;

procedure DelDoublettes(var FMP:tFacMulPow);
//throw out doublettes,
var
pNext,pCurrent : tpFac_mul;
i, len,idx : integer;
begin
len := 0;
pCurrent := @FMP[0];
pNext := pCurrent;
For i := 0 to High(FMP)-1 do
begin
inc(pNext);
// don't increment pCurrent if equal
// pCurrent gets or stays the highest Value in n!^high_pow
if pCurrent^.fm_num = pNext^.fm_num then
Begin
idx := pCurrent^.fm_high_idx;
if idx < pNext^.fm_high_idx then
pCurrent^  := pNext^
else
if idx = pNext^.fm_high_idx then
if pCurrent^.fm_high_pow < pNext^.fm_high_pow then
pCurrent^  := pNext^;
end
else
begin
inc(len);
inc(pCurrent);
pCurrent^  := pNext^;
end;
end;
setlength(FMP,len);
end;

procedure QuickSort(var AI: tFacMulPow; ALo, AHi: Int32);
var
Tmp :tFac_mul;
Pivot : tnum;
Lo, Hi : Int32;
begin
Lo := ALo;
Hi := AHi;
Pivot := AI[(Lo + Hi) div 2].fm_num;
repeat
while AI[Lo].fm_num < Pivot do
Inc(Lo);
while AI[Hi].fm_num > Pivot do
Dec(Hi);
if Lo <= Hi then
begin
Tmp := AI[Lo];
AI[Lo] := AI[Hi];
AI[Hi] := Tmp;
Inc(Lo);
Dec(Hi);
end;
until Lo > Hi;
if Hi > ALo then
QuickSort(AI, ALo, Hi) ;
if Lo < AHi then
QuickSort(AI, Lo, AHi) ;
end;

function InsertFacMulPow(var res:tFacMulPow;Facidx:integer):boolean;
var
Fac,FacPow,NewNum,limit : tnum;
l_res,l_NewMaxPow,idx,i,j : Integer;
begin
fac := Factorial[Facidx];
if fac>dblLimit then
EXIT(false);

if length(res)> 0 then
begin
l_NewMaxPow := trunc(ln(dblLimit)/ln(Fac))+1;
l_res := length(res);
//calc new length, reduces allocation of big memory chunks
//first original length + length of the new to insert
j := l_res+l_NewMaxPow;
//find the maximal needed elements which stay below  dbllimit
// for every Fac^i
idx := High(res);
FacPow := Fac;
For i := 1 to l_NewMaxPow do
Begin
limit := dblLimit DIV FacPow;
if limit < 1 then
BREAK;
//search for the right position
repeat
dec(idx);
until res[idx].fm_num<=limit;
inc(j,idx);
FacPow *= fac;
end;
j += 2;
setlength(res,j);

idx := l_res;
FacPow := fac;
For j := 1 to l_NewMaxPow do
begin
For i := 0 to l_res do
begin
NewNum := res[i].fm_num*FacPow;
if NewNum>dblLimit then
Break;
res[idx]:= res[i];
with res[idx] do
Begin
fm_num := NewNum;
include(fm_pow,Facidx);
fm_high_idx := Facidx;
fm_high_pow := j;
end;
inc(idx);
end;
FacPow *= fac;
end;
setlength(res,idx);
QuickSort(res,Low(res),High(res));
DelDoublettes(res);
end
else
GenerateFirst(Facidx,res);
Exit(true);
end;

var
i : integer;
BEGIn
InitFirst;

repeat
ResetSearch;
i := 0;
repeat
if Not(InsertFacMulPow(FacMulPowGes,i)) then
BREAK;
inc(i);
until i > High(Factorial);
//check if MaxIdx is found
if (Length(FacMulPowGes) > MaxIdx) then
begin
if (LastSearchFor.fm_num<> FacMulPowGes[MaxIdx-1].fm_num) then
Begin
LastSearchFor := FacMulPowGes[MaxIdx-1];
//the next factorial is to big, so search is done
if LastSearchFor.fm_num < Factorial[i] then
break;
end
else
Break;
end;
if dblLimit> HIGH(tNUm) DIV 256 then
BREAK;
dblLimit *= 256;
until false;

write('Found ',length(FacMulPowGes),' Jordan-Polia numbers ');
writeln('up to ',CommatizeUint64(dblLimit));
writeln;

Out_First_N(50);

write('The last < 1E8 ');
for i := 0 to High(FacMulPowGes) do
if FacMulPowGes[i].fm_num > 1E8 then
begin
Out_MulFac(i,FacMulPowGes[i-1]);
BREAK;
end;
writeln;

Out_I_th(1);
Out_I_th(100);
Out_I_th(800);
Out_I_th(1050);
Out_I_th(1800);
Out_I_th(2800);
Out_I_th(3800);
END.
```
@home:
```Found 3876 Jordan-Polia numbers up to 9,007,199,254,740,992

The first 50 Jordan-Polia numbers
1     2     4     6     8    12    16    24    32    36    48    64
72    96   120   128   144   192   216   240   256   288   384   432
480   512   576   720   768   864   960 1,024 1,152 1,296 1,440 1,536
1,728 1,920 2,048 2,304 2,592 2,880 3,072 3,456 3,840 4,096 4,320 4,608
5,040 5,184

The last < 1E8                99,532,800 =  (6!)^2 4! (2!)^3

1-th :                         1 =  1
100-th :                    92,160 =  6! (2!)^7
800-th :            18,345,885,696 =  (4!)^7 (2!)^2
1050-th :           139,345,920,000 =  8! (5!)^3 2!
1800-th :         9,784,472,371,200 =  (6!)^2 (4!)^2 (2!)^15
2800-th :       439,378,587,648,000 =  14! 7!
3800-th :     7,213,895,789,838,336 =  (4!)^8 (2!)^16
real	0m0,004s user	0m0,004s sys	0m0,000s```

## Perl

Translation of: Raku
Library: ntheory
```use strict;
use warnings;
use feature 'say';

use ntheory 'factorial';
use List::AllUtils <max firstidx>;

sub table { my \$t = 10 * (my \$c = 1 + length max @_); ( sprintf( ('%'.\$c.'d')x@_, @_) ) =~ s/.{1,\$t}\K/\n/gr }

sub Jordan_Polya {
my \$limit = shift;
my(\$k,@JP) = (2);
push @JP, factorial \$_ for 0..18;

while (\$k < @JP) {
my \$rk = \$JP[\$k];
for my \$l (2 .. @JP) {
my \$kl = \$JP[\$l] * \$rk;
last if \$kl > \$limit;
LOOP: {
my \$p = firstidx { \$_ >= \$kl } @JP;
if    (\$p  < \$#JP and \$JP[\$p] != \$kl) { splice @JP, \$p, 0, \$kl }
elsif (\$p == \$#JP                   ) {   push @JP,        \$kl }
\$kl > \$limit/\$rk ? last LOOP : (\$kl *= \$rk)
}
}
\$k++
}
shift @JP; return @JP
}

my @JP = Jordan_Polya 2**27;
say "First 50 Jordan-Pólya numbers:\n" . table @JP[0..49];
say 'The largest Jordan-Pólya number before 100 million: ' . \$JP[-1 + firstidx { \$_ > 1e8 } @JP];
```
Output:
```First 50 Jordan-Pólya numbers:
1    2    4    6    8   12   16   24   32   36
48   64   72   96  120  128  144  192  216  240
256  288  384  432  480  512  576  720  768  864
960 1024 1152 1296 1440 1536 1728 1920 2048 2304
2592 2880 3072 3456 3840 4096 4320 4608 5040 5184

The largest Jordan-Pólya number before 100 million: 99532800
```

## Phix

```with javascript_semantics
function factorials_le(atom limit)
sequence res = {}
while true do
atom nf = factorial(length(res)+1)
if nf>limit then exit end if
res &= nf
end while
return res
end function

function jp(atom limit)
sequence res = factorials_le(limit)
integer k=2
while k<=length(res) do
atom rk = res[k]
for l=2 to length(res) do
atom kl = res[l]*rk
if kl>limit then exit end if
do
integer p = binary_search(kl,res)
if p<0 then
p = abs(p)
res = res[1..p-1] & kl & res[p..\$]
end if
kl *= rk
until kl>limit
end for
k += 1
end while
return res
end function

function decompose(atom jp)
--
-- Subtract prime powers of factorials off the prime powers of the jp number,
-- only for factorials that have the same high prime factor as the remainder,
-- and only putting things back on the todo list if still viable.
-- Somewhat slowish, but at least it /is/ very thorough.
--
sequence p = prime_powers(jp)
integer lp = length(p),
mp = p[lp][1],  -- (max prime factor)
hf = get_prime(lp+1)-1 -- (high factorial)
assert(mp = get_prime(lp))
sequence ap = get_primes_le(mp), -- (all primes)
fs = apply(tagset(hf),factorial),
fp = apply(fs,prime_powers),
pf = repeat(0,hf), -- (powers of factorials)
todo = {{p,pf}},
seen = {},
result = {}
while length(todo) do
{{p,pf},todo} = {todo[1],todo[2..\$]}
for fdx,fpi in fp from 2 do
if fpi[\$][1] = p[\$][1] then -- same max prime factor
bool ok = true
for j,fpij in fpi do
if fpij[2]>p[find(fpij[1],ap)][2] then
-- this factorial ain't a factor
ok = false
exit
end if
end for
if ok then
-- reduce & trim the remaining prime powers:
sequence pnxt = deep_copy(p)
for j,fpij in fpi do
pnxt[find(fpij[1],ap)][2] -= fpij[2]
end for
while length(pnxt) and pnxt[\$][2]=0 do
pnxt = pnxt[1..\$-1]
end while
sequence fnxt = deep_copy(pf)
fnxt[fdx] += 1 -- **one** extra factorial power
if length(pnxt) then
for i=2 to length(pnxt) do
if pnxt[i][2]>pnxt[i-1][2] then
-- ie/eg you cannot ever knock a 7! or above off
--  if there ain't enough 5 (and 3 and 2) avail.
exit
end if
end for
and not find({pnxt,fnxt},seen) then
seen = append(seen,{pnxt,fnxt})
todo = append(todo,{pnxt,fnxt})
end if
else
result = append(result,fnxt)
end if
end if
end if
end for
end while
result = reverse(sort(apply(result,reverse))[\$])
string res = ""
for i=length(result) to 1 by -1 do
if result[i] then
if length(res) then res &= " * " end if
res &= sprintf("%d!",i)
if result[i]>1 then
res &= sprintf("^%d",result[i])
end if
end if
end for
return res
end function

atom t0 = time()
sequence r = jp(power(2,53)-1)
printf(1,"%d Jordan-Polya numbers found, the first 50 are:\n%s\n",
{length(r),join_by(r[1..50],1,10," ",fmt:="%4d")})
printf(1,"The largest under 100 million: %,d\n",r[abs(binary_search(1e8,r))-1])
for i in {100,800,1050,1800,2800,3800} do
printf(1,"The %d%s is %,d = %s\n",{i,ord(i),r[i],decompose(r[i])})
end for
?elapsed(time()-t0)
```
Output:
```3887 Jordan-Polya numbers found, the first 50 are:
1    2    4    6    8   12   16   24   32   36
48   64   72   96  120  128  144  192  216  240
256  288  384  432  480  512  576  720  768  864
960 1024 1152 1296 1440 1536 1728 1920 2048 2304
2592 2880 3072 3456 3840 4096 4320 4608 5040 5184

The largest under 100 million: 99,532,800
The 100th is 92,160 = 6! * 2!^7
The 800th is 18,345,885,696 = 4!^7 * 2!^2
The 1050th is 139,345,920,000 = 8! * 5!^3 * 2!
The 1800th is 9,784,472,371,200 = 6!^2 * 4!^2 * 2!^15
The 2800th is 439,378,587,648,000 = 14! * 7!
The 3800th is 7,213,895,789,838,336 = 4!^8 * 2!^16
"1.5s"
```

Some 80%-90% of the time is now spent in the decomposing phase.

## Raku

Partial translation of Go

```# 20230719 Raku programming solution

my \factorials = 1, | [\*] 1..18; # with 0!

sub JordanPolya (\limit) {
my \ix = (factorials.keys.first: factorials[*] >= limit) // factorials.end;
my (\$k, @res) = 2, |factorials[0..ix];

while \$k < @res.elems {
my \rk = @res[\$k];
for 2 .. @res.elems -> \l {
my \kl = \$ = @res[l] * rk;
last if kl > limit;
loop {
my \p = @res.keys.first: { @res[\$_] >= kl } # performance
if p < @res.elems and @res[p] != kl {
@res.splice: p, 0, kl
} elsif p == @res.elems {
@res.append: kl
}
kl > limit/rk ?? ( last ) !! kl *= rk
}
}
\$k++
}
return @res[1..*]
}

my @result = JordanPolya 2**30 ;
say "First 50 Jordan-Pólya numbers:";
say [~] \$_>>.fmt('%5s') for @result[^50].rotor(10);
print "\nThe largest Jordan-Pólya number before 100 million: ";
say @result.first: * < 100_000_000, :end;
```

You may Attempt This Online!

## Scala

Translation of: Java
```import java.util.{ArrayList, HashMap, TreeMap, TreeSet}
import scala.jdk.CollectionConverters._

object JordanPolyaNumbers {

private val jordanPolyaSet = new TreeSet[Long]()
private val decompositions = new HashMap[Long, TreeMap[Integer, Integer]]()

def main(args: Array[String]): Unit = {
createJordanPolya()

val belowHundredMillion = jordanPolyaSet.floor(100_000_000L)
val jordanPolya = new ArrayList[Long](jordanPolyaSet)

println("The first 50 Jordan-Polya numbers:")
jordanPolya.asScala.take(50).zipWithIndex.foreach { case (number, index) =>
print(f"\$number%5s\${if(index % 10 == 9) "\n" else ""}")
}
println()

println(s"The largest Jordan-Polya number less than 100 million: \$belowHundredMillion")
println()

List(800, 1050, 1800, 2800, 3800).foreach { i =>
println(s"The \${i}th Jordan-Polya number is: \${jordanPolya.get(i - 1)} = \${toString(decompositions.get(jordanPolya.get(i - 1)))}")
}
}

private def createJordanPolya(): Unit = {
val nextSet = new TreeSet[Long]()
decompositions.put(1L, new TreeMap[Integer, Integer]())
var factorial = 1L

for (multiplier <- 2 to 20) {
factorial *= multiplier
val it = jordanPolyaSet.iterator()
while (it.hasNext) {
var number = it.next()
while (number <= Long.MaxValue / factorial) {
val original = number
number *= factorial
decompositions.put(number, new TreeMap[Integer, Integer](decompositions.get(original)))
val currentMap = decompositions.get(number)
currentMap.merge(multiplier, 1, (a: Integer, b: Integer) => Integer.sum(a, b))
}
}
nextSet.clear()
}
}

private def toString(aMap: TreeMap[Integer, Integer]): String = {
aMap.descendingMap().asScala.map { case (key, value) =>
s"\$key!\${if (value == 1) "" else "^" + value}"
}.mkString(" * ")
}
}
```
Output:
```The first 50 Jordan-Polya numbers:
1    2    4    6    8   12   16   24   32   36
48   64   72   96  120  128  144  192  216  240
256  288  384  432  480  512  576  720  768  864
960 1024 1152 1296 1440 1536 1728 1920 2048 2304
2592 2880 3072 3456 3840 4096 4320 4608 5040 5184

The largest Jordan-Polya number less than 100 million: 99532800

The 800th Jordan-Polya number is: 18345885696 = 4!^7 * 2!^2
The 1050th Jordan-Polya number is: 139345920000 = 8! * 5!^3 * 2!
The 1800th Jordan-Polya number is: 9784472371200 = 6!^2 * 4!^2 * 2!^15
The 2800th Jordan-Polya number is: 439378587648000 = 14! * 7!
The 3800th Jordan-Polya number is: 7213895789838336 = 4!^8 * 2!^16
```

You may Attempt This Online!

## Wren

### Version 1

Library: Wren-set
Library: Wren-seq
Library: Wren-fmt

This uses the recursive PARI/Python algorithm in the OEIS entry.

```import "./set" for Set
import "./seq" for Lst
import "./fmt" for Fmt

var JordanPolya = Fn.new { |lim, mx|
if (lim < 2) return [1]
var v = Set.new()
var t = 1
if (!mx) mx = lim
for (k in 2..mx) {
t = t * k
if (t > lim) break
for (rest in JordanPolya.call((lim/t).floor, t)) {
}
}
return v.toList.sort()
}

var factorials = List.filled(19, 1)

var cacheFactorials = Fn.new {
var fact = 1
for (i in 2..18) {
fact = fact * i
factorials[i] = fact
}
}

var Decompose = Fn.new { |n, start|
if (!start) start = 18
if (start < 2) return []
var m = n
var f = []
while (m % factorials[start] == 0) {
m =  m / factorials[start]
if (m == 1) return f
}
if (f.count > 0) {
var g = Decompose.call(m, start-1)
if (g.count > 0) {
var prod = 1
for (e in g) prod = prod * factorials[e]
if (prod == m) return f + g
}
}
return Decompose.call(n, start-1)
}

cacheFactorials.call()
var v = JordanPolya.call(2.pow(53)-1, null)
System.print("First 50 Jordan–Pólya numbers:")
Fmt.tprint("\$4d ", v[0..49], 10)

System.write("\nThe largest Jordan–Pólya number before 100 million: ")
for (i in 1...v.count) {
if (v[i] > 1e8) {
Fmt.print("\$,d\n", v[i-1])
break
}
}

for (i in [800, 1050, 1800, 2800, 3800]) {
Fmt.print("The \$,r Jordan-Pólya number is : \$,d", i, v[i-1])
var g = Lst.individuals(Decompose.call(v[i-1], null))
var s = g.map { |l|
if (l[1] == 1) return "%(l[0])!"
return Fmt.swrite("(\$d!)\$S", l[0], l[1])
}.join(" x ")
Fmt.print("= \$s\n", s)
}
```
Output:
```First 50 Jordan–Pólya numbers:
1     2     4     6     8    12    16    24    32    36
48    64    72    96   120   128   144   192   216   240
256   288   384   432   480   512   576   720   768   864
960  1024  1152  1296  1440  1536  1728  1920  2048  2304
2592  2880  3072  3456  3840  4096  4320  4608  5040  5184

The largest Jordan–Pólya number before 100 million: 99,532,800

The 800th Jordan-Pólya number is : 18,345,885,696
= (4!)⁷ x (2!)²

The 1,050th Jordan-Pólya number is : 139,345,920,000
= 8! x (5!)³ x 2!

The 1,800th Jordan-Pólya number is : 9,784,472,371,200
= (6!)² x (4!)² x (2!)¹⁵

The 2,800th Jordan-Pólya number is : 439,378,587,648,000
= 14! x 7!

The 3,800th Jordan-Pólya number is : 7,213,895,789,838,336
= (4!)⁸ x (2!)¹⁶
```

### Version 2

Library: Wren-sort

This uses the same non-recursive algorithm as the Phix entry to generate the J-P numbers which, at 1.1 seconds on my machine, is about 40 times quicker than the OEIS algorithm.

```import "./sort" for Find
import "./seq" for Lst
import "./fmt" for Fmt

var factorials = List.filled(19, 1)

var cacheFactorials = Fn.new {
var fact = 1
for (i in 2..18) {
fact = fact * i
factorials[i] = fact
}
}

var JordanPolya = Fn.new { |limit|
var ix = Find.nearest(factorials, limit).min(18)
var res = factorials[0..ix]
var k = 2
while (k < res.count) {
var rk = res[k]
for (l in 2...res.count) {
var kl = res[l] * rk
if (kl > limit) break
while (true) {
var p = Find.nearest(res, kl)
if (p < res.count && res[p] != kl) {
res.insert(p, kl)
} else if (p == res.count) {
}
kl = kl * rk
if (kl > limit) break
}
}
k = k + 1
}
return res[1..-1]
}

var Decompose = Fn.new { |n, start|
if (!start) start = 18
if (start < 2) return []
var m = n
var f = []
while (m % factorials[start] == 0) {
m =  m / factorials[start]
if (m == 1) return f
}
if (f.count > 0) {
var g = Decompose.call(m, start-1)
if (g.count > 0) {
var prod = 1
for (e in g) prod = prod * factorials[e]
if (prod == m) return f + g
}
}
return Decompose.call(n, start-1)
}

cacheFactorials.call()
var v = JordanPolya.call(2.pow(53)-1)
System.print("First 50 Jordan–Pólya numbers:")
Fmt.tprint("\$4d ", v[0..49], 10)

System.write("\nThe largest Jordan–Pólya number before 100 million: ")
for (i in 1...v.count) {
if (v[i] > 1e8) {
Fmt.print("\$,d\n", v[i-1])
break
}
}

for (i in [800, 1050, 1800, 2800, 3800]) {
Fmt.print("The \$,r Jordan-Pólya number is : \$,d", i, v[i-1])
var g = Lst.individuals(Decompose.call(v[i-1], null))
var s = g.map { |l|
if (l[1] == 1) return "%(l[0])!"
return Fmt.swrite("(\$d!)\$S", l[0], l[1])
}.join(" x ")
Fmt.print("= \$s\n", s)
}
```
Output:
```Identical to first version.
```

## XPL0

Simple-minded brute force. 20 seconds on Pi4. No bonus.

```int Factorials(1+12);

func IsJPNum(N0);
int  N0;
int  N, Limit, I, Q;
[Limit:= 7;
N:= N0;
loop    [I:= Limit;
loop    [Q:= N / Factorials(I);
if rem(0) = 0 then
[if Q = 1 then return true;
N:= Q;
]
else    I:= I-1;
if I = 1 then
[if Limit = 1 then return false;
Limit:= Limit-1;
N:= N0;
quit;
]
];
];
];

int F, N, C, SN;
[F:= 1;
for N:= 1 to 12 do
[F:= F*N;
Factorials(N):= F;
];
Text(0, "First 50 Jordan-Polya numbers:^m^j");
Format(5, 0);
RlOut(0, 1.);           \handle odd number exception
C:= 1;  N:= 2;
loop    [if IsJPNum(N) then
[C:= C+1;
if C <= 50 then
[RlOut(0, float(N));
if rem(C/10) = 0 then CrLf(0);
];
SN:= N;
];
N:= N+2;
if N >= 100_000_000 then quit;
];
Text(0, "^m^jThe largest Jordan-Polya number before 100 million: ");
IntOut(0, SN);  CrLf(0);
]```
Output:
```First 50 Jordan-Polya numbers:
1    2    4    6    8   12   16   24   32   36
48   64   72   96  120  128  144  192  216  240
256  288  384  432  480  512  576  720  768  864
960 1024 1152 1296 1440 1536 1728 1920 2048 2304
2592 2880 3072 3456 3840 4096 4320 4608 5040 5184

The largest Jordan-Polya number before 100 million: 99532800
```