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Ingy döt Net 2023-07-01 11:58:00 -04:00
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5
Task/Partition-function-P/00-META.yaml Normal file
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---
category:
- Memoization
from: http://rosettacode.org/wiki/Partition_function_P
note: Recursion

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Task/Partition-function-P/00-TASK.txt Normal file
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The [https://mathworld.wolfram.com/PartitionFunctionP.html Partition Function P] is the function P(n), where n∈ℤ, defined as the number of distinct ways in which n can be expressed as the sum of non-increasing positive integers.
;Example:
P(4) = 5 because 4 = <big>Σ</big>(4) = <big>Σ</big>(3,1) = <big>Σ</big>(2,2) = <big>Σ</big>(2,1,1) = <big>Σ</big>(1,1,1,1)
P(n) can be expressed as the recurrence relation:
P(n) = P(n-1) +P(n-2) -P(n-5) -P(n-7) +P(n-12) +P(n-15) -P(n-22) -P(n-26) +P(n-35) +P(n-40) ...
The successive numbers in the above equation have the differences: &nbsp; 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15, 8 ...
This task may be of popular interest because [https://www.youtube.com/channel/UC1_uAIS3r8Vu6JjXWvastJg Mathologer] made the video, [https://www.youtube.com/watch?v=iJ8pnCO0nTY The hardest "What comes next?" (Euler's pentagonal formula)], where he asks the programmers among his viewers to calculate P(666). The video was viewed more than 100,000 times in the first couple of weeks after its release.
In Wolfram Language, this function has been implemented as PartitionsP.
;Task:
Write a function which returns the value of PartitionsP(n). Solutions can be iterative or recursive.
Bonus task: show how long it takes to compute PartitionsP(6666).
;References:
* [https://www.youtube.com/watch?v=iJ8pnCO0nTY The hardest "What comes next?" (Euler's pentagonal formula)] The explanatory video by Mathologer that makes this task a popular interest.
* [https://mathworld.wolfram.com/PartitionFunctionP.html Partition Function P] Mathworld entry for the Partition function.
* [[wp:Partition_function_(number_theory)|Partition function (number theory)]] Wikipedia entry for the Partition function.
;Related tasks:
* [[9 billion names of God the integer]]
<br><br>

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Task/Partition-function-P/11l/partition-function-p.11l Normal file
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F partitions(n)
V p = [BigInt(1)] [+] [BigInt(0)] * n
L(i) 1 .. n
V k = 0
L
k++
V j = (k * (3 * k - 1)) I/ 2
I j > i
L.break
I k [&] 1
p[i] += p[i - j]
E
p[i] -= p[i - j]
j = (k * (3 * k + 1)) I/ 2
I j > i
L.break
I k [&] 1
p[i] += p[i - j]
E
p[i] -= p[i - j]
R p[n]
print(‘Partitions: ’(0.<15).map(x -> partitions(x)))
V start = time:perf_counter()
print(partitions(6666))
print(time:perf_counter() - start)

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Task/Partition-function-P/C++/partition-function-p-1.cpp Normal file
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#include <chrono>
#include <iostream>
#include <vector>
#include <gmpxx.h>
using big_int = mpz_class;
big_int partitions(int n) {
std::vector<big_int> p(n + 1);
p[0] = 1;
for (int i = 1; i <= n; ++i) {
for (int k = 1;; ++k) {
int j = (k * (3*k - 1))/2;
if (j > i)
break;
if (k & 1)
p[i] += p[i - j];
else
p[i] -= p[i - j];
j = (k * (3*k + 1))/2;
if (j > i)
break;
if (k & 1)
p[i] += p[i - j];
else
p[i] -= p[i - j];
}
}
return p[n];
}
int main() {
auto start = std::chrono::steady_clock::now();
auto result = partitions(6666);
auto end = std::chrono::steady_clock::now();
std::chrono::duration<double, std::milli> ms(end - start);
std::cout << result << '\n';
std::cout << "elapsed time: " << ms.count() << " milliseconds\n";
}

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Task/Partition-function-P/C++/partition-function-p-2.cpp Normal file
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#include <chrono>
#include <iostream>
using namespace std;
using namespace chrono;
const long long Lm = (long)1e18;
const int Fm = 18;
struct LI { long long lo, ml, mh, hi, tp; };
LI set(long long s) { LI d;
d.lo = s; d.ml = d.mh = d.hi = d.tp = 0; return d; }
void inc(LI& d, LI s) { // d += s
if ((d.lo += s.lo) >= Lm) { d.ml++; d.lo -= Lm; }
if ((d.ml += s.ml) >= Lm) { d.mh++; d.ml -= Lm; }
if ((d.mh += s.mh) >= Lm) { d.hi++; d.mh -= Lm; }
if ((d.hi += s.hi) >= Lm) { d.tp++; d.hi -= Lm; }
d.tp += s.tp;
}
void dec(LI& d, LI s) { // d -= s
if ((d.lo -= s.lo) < 0) { d.ml--; d.lo += Lm; }
if ((d.ml -= s.ml) < 0) { d.mh--; d.ml += Lm; }
if ((d.mh -= s.mh) < 0) { d.hi--; d.mh += Lm; }
if ((d.hi -= s.hi) < 0) { d.tp--; d.hi += Lm; }
d.tp -= s.tp;
}
inline string sf(long long n) {
int len = Fm;
string result(len--, '0');
for (long long i = n; len >= 0 && i > 0; --len, i /= 10)
result[len] = '0' + i % 10;
return result;
}
string fmt(LI x) { // returns formatted string value of x
if (x.tp > 0) return to_string(x.tp) + sf(x.hi) + sf(x.mh) + sf(x.ml) + sf(x.lo);
if (x.hi > 0) return to_string(x.hi) + sf(x.mh) + sf(x.ml) + sf(x.lo);
if (x.mh > 0) return to_string(x.mh) + sf(x.ml) + sf(x.lo);
if (x.ml > 0) return to_string(x.ml) + sf(x.lo);
return to_string(x.lo);
}
LI partcount(int n) { LI p[n + 1]; p[0] = set(1);
for (int i = 1; i <= n; i++) {
int k = 0, d = -2, j = i; LI x = set(0); while (true) {
if ((j -= (d += 3) - k) >= 0) inc(x, p[j]); else break;
if ((j -= ++k) >= 0) inc(x, p[j]); else break;
if ((j -= (d += 3) - k) >= 0) dec(x, p[j]); else break;
if ((j -= ++k) >= 0) dec(x, p[j]); else break;
} p[i] = x; } return p[n]; }
int main() {
auto start = steady_clock::now();
auto result = partcount(6666);
auto end = steady_clock::now();
duration<double, milli> ms(end - start);
cout << fmt(result) << " " << ms.count() << " ms";
}

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Task/Partition-function-P/C-sharp/partition-function-p.cs Normal file
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using System;
class Program {
const long Lm = (long)1e18;
const string Fm = "D18";
// provides 5 x 18 = 90 decimal digits
struct LI { public long lo, ml, mh, hi, tp; }
static void inc(ref LI d, LI s) { // d += s
if ((d.lo += s.lo) >= Lm) { d.ml++; d.lo -= Lm; }
if ((d.ml += s.ml) >= Lm) { d.mh++; d.ml -= Lm; }
if ((d.mh += s.mh) >= Lm) { d.hi++; d.mh -= Lm; }
if ((d.hi += s.hi) >= Lm) { d.tp++; d.hi -= Lm; }
d.tp += s.tp;
}
static void dec(ref LI d, LI s) { // d -= s
if ((d.lo -= s.lo) < 0) { d.ml--; d.lo += Lm; }
if ((d.ml -= s.ml) < 0) { d.mh--; d.ml += Lm; }
if ((d.mh -= s.mh) < 0) { d.hi--; d.mh += Lm; }
if ((d.hi -= s.hi) < 0) { d.tp--; d.hi += Lm; }
d.tp -= s.tp;
}
static LI set(long s) { LI d;
d.lo = s; d.ml = d.mh = d.hi = d.tp = 0; return d; }
static string fmt(LI x) { // returns formatted string value of x
if (x.tp > 0) return x.tp.ToString() + x.hi.ToString(Fm) + x.mh.ToString(Fm) + x.ml.ToString(Fm) + x.lo.ToString(Fm);
if (x.hi > 0) return x.hi.ToString() + x.mh.ToString(Fm) + x.ml.ToString(Fm) + x.lo.ToString(Fm);
if (x.mh > 0) return x.mh.ToString() + x.ml.ToString(Fm) + x.lo.ToString(Fm);
if (x.ml > 0) return x.ml.ToString() + x.lo.ToString(Fm);
return x.lo.ToString();
}
static LI partcount(int n) {
var P = new LI[n + 1]; P[0] = set(1);
for (int i = 1; i <= n; i++) {
int k = 0, d = -2, j = i;
LI x = set(0);
while (true) {
if ((j -= (d += 3) -k) >= 0) inc(ref x, P[j]); else break;
if ((j -= ++k) >= 0) inc(ref x, P[j]); else break;
if ((j -= (d += 3) -k) >= 0) dec(ref x, P[j]); else break;
if ((j -= ++k) >= 0) dec(ref x, P[j]); else break;
}
P[i] = x;
}
return P[n];
}
static void Main(string[] args) {
var sw = System.Diagnostics.Stopwatch.StartNew ();
var res = partcount(6666); sw.Stop();
Console.Write("{0} {1} ms", fmt(res), sw.Elapsed.TotalMilliseconds);
}
}

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Task/Partition-function-P/C/partition-function-p.c Normal file
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#include <stdint.h>
#include <stdlib.h>
#include <stdio.h>
#include <time.h>
#include <gmp.h>
mpz_t* partition(uint64_t n) {
mpz_t *pn = (mpz_t *)malloc((n + 2) * sizeof(mpz_t));
mpz_init_set_ui(pn[0], 1);
mpz_init_set_ui(pn[1], 1);
for (uint64_t i = 2; i < n + 2; i ++) {
mpz_init(pn[i]);
for (uint64_t k = 1, penta; ; k++) {
penta = k * (3 * k - 1) >> 1;
if (penta >= i) break;
if (k & 1) mpz_add(pn[i], pn[i], pn[i - penta]);
else mpz_sub(pn[i], pn[i], pn[i - penta]);
penta += k;
if (penta >= i) break;
if (k & 1) mpz_add(pn[i], pn[i], pn[i - penta]);
else mpz_sub(pn[i], pn[i], pn[i - penta]);
}
}
mpz_t *tmp = &pn[n + 1];
for (uint64_t i = 0; i < n + 1; i ++) mpz_clear(pn[i]);
free(pn);
return tmp;
}
int main(int argc, char const *argv[]) {
clock_t start = clock();
mpz_t *p = partition(6666);
gmp_printf("%Zd\n", p);
printf("Elapsed time: %.04f seconds\n",
(double)(clock() - start) / (double)CLOCKS_PER_SEC);
return 0;
}

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Task/Partition-function-P/Delphi/partition-function-p.delphi Normal file
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program Partition_function_P;
{$APPTYPE CONSOLE}
uses
System.SysUtils,
Velthuis.BigIntegers,
System.Diagnostics;
var
p: TArray<BigInteger>;
pd: TArray<Integer>;
function PartiDiffDiff(n: Integer): Integer;
begin
if n and 1 = 1 then
exit((n + 1) div 2);
Result := n + 1;
end;
function partDiff(n: Integer): Integer;
begin
if n < 2 then
exit(1);
pd[n] := pd[n - 1] + PartiDiffDiff(n - 1);
Result := pd[n];
end;
procedure partitionP(n: Integer);
begin
if n < 2 then
exit;
var psum: BigInteger := 0;
for var i := 1 to n do
begin
var pdi := partDiff(i);
if pdi > n then
Break;
var sign: Int64 := -1;
if (i - 1) mod 4 < 2 then
sign := 1;
var t: BigInteger := BigInteger(p[n - pdi]) * BigInteger(sign);
psum := psum + t;
end;
p[n] := psum;
end;
begin
var stopwatch := TStopwatch.Create;
const n = 6666;
SetLength(p, n + 1);
SetLength(pd, n + 1);
stopwatch.Start;
p[0] := 1;
pd[0] := 1;
p[1] := 1;
pd[1] := 1;
for var i := 2 to n do
partitionP(i);
stopwatch.Stop;
writeln(format('p[%d] = %s', [n, p[n].ToString]));
writeln('Took ', stopwatch.ElapsedMilliseconds, 'ms');
Readln;
end.

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Task/Partition-function-P/Elixir/partition-function-p.elixir Normal file
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use Bitwise, skip_operators: true
defmodule Partition do
def init(), do:
:ets.new :pN, [:set, :named_table, :private]
def gpentagonals(), do: Stream.unfold {1, 0}, &next/1
defp next({m, n}) do
a = case rem m, 2 do
0 -> div m, 2
1 -> m
end
{n, {m + 1, n + a}}
end
def p(0), do: 1
def p(n) do
case :ets.lookup :pN, n do
[{^n, val}] -> val
[] ->
{val, _} = gpentagonals()
|> Stream.drop(1)
|> Stream.take_while(fn m -> m <= n end)
|> Stream.map(fn g -> p(n - g) end)
|> Enum.reduce({0, 0},
fn n, {a, sgn} -> {
a + (if sgn < 2, do: n, else: -n),
band(sgn + 1, 3)
}
end)
:ets.insert :pN, {n, val}
val
end
end
end
Partition.init
IO.puts Partition.p 6666

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Task/Partition-function-P/Erlang/partition-function-p.erl Normal file
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-mode(compile).
main(_) ->
ets:new(pN, [set, named_table, protected]),
io:format("~w~n", [p(6666)]).
p(0) -> 1;
p(N) ->
case ets:lookup(pN, N) of
[{N, Pn}] -> Pn;
[] ->
Terms = [p(N - G) || G <- gpentagonals(N)],
Pn = sum_partitions(Terms),
ets:insert(pN, {N, Pn}),
Pn
end.
sum_partitions(Terms) -> sum_partitions(Terms, 0, 0).
sum_partitions([], _, Sum) -> Sum;
sum_partitions([N|Ns], Sgn, Sum) ->
Summand = case Sgn < 2 of
true -> N;
false -> -N
end,
sum_partitions(Ns, (Sgn+1) band 3, Sum + Summand).
gpentagonals(Max) -> gpentagonals(1, Max, [0]).
gpentagonals(M, Max, Ps = [N|_]) ->
GP = N + case M rem 2 of
0 -> M div 2;
1 -> M
end,
if
GP > Max -> tl(lists:reverse(Ps));
true -> gpentagonals(M + 1, Max, [GP|Ps])
end.

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Task/Partition-function-P/F-Sharp/partition-function-p.fs Normal file
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// PartionP: Nigel Galloway. April 12th., 2021
let pP g=let rec fN i g e l=seq{yield(l,e+i);yield(-l,e+i+g);yield! fN(i+1)(g+2)(e+i+g)(-l)}
let N,G=Array.create(g+1) 1I,seq{yield (1I,1);yield! fN 1 3 1 1I}|>Seq.takeWhile(fun(_,n)->n<=g)|>List.ofSeq
seq{2..g}|>Seq.iter(fun p->N.[p]<-G|>List.takeWhile(fun(_,n)->n<=p)|>Seq.fold(fun Σ (n,g)->Σ+n*N.[p-g]) 0I); N.[g]
printfn "666->%A\n\n6666->%A\n\n123456->%A" (pP 666) (pP 6666) (pP 123456)

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Task/Partition-function-P/Factor/partition-function-p.factor Normal file
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USING: kernel lists lists.lazy math sequences sequences.extras ;
! Compute the nth pentagonal number.
: penta ( n -- m ) [ sq 3 * ] [ - 2/ ] bi ;
! An infinite lazy list of indices to add and subtract in the
! sequence of partitions to find the next P.
: seq ( -- list )
1 lfrom [ penta 1 - ] <lazy-map> 1 lfrom [ neg penta 1 - ]
<lazy-map> lmerge ;
! Reduce a sequence by adding two, subtracting two, adding two,
! etc...
: ++-- ( seq -- n ) 0 [ 2/ odd? [ - ] [ + ] if ] reduce-index ;
! Find the next member of the partitions sequence.
: step ( seq pseq -- seq 'pseq )
dup length [ < ] curry pick swap take-while over <reversed>
nths ++-- suffix! ;
: partitions ( m -- n )
[ seq swap [ <= ] curry lwhile list>array ]
[ V{ 1 } clone swap [ step ] times last nip ] bi ;

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Task/Partition-function-P/FreeBASIC/partition-function-p-1.basic Normal file
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Function PartitionsP(n As UInteger) As ULongInt
' if n > 416, the result becomes to large for a unsigned 64bit integer
Dim As ULongInt p(n)
Dim As UInteger k, j
p(0) = 1
For i As UInteger = 1 To n
k = 0
While TRUE
k += 1
j = (k * (3*k - 1)) \ 2
If (j > i) Then Exit While
If (k And 1) Then
p(i) += p(i - j)
Else
p(i) -= p(i - j)
End If
'j = (k * (3*k + 1)) \ 2
j += k
If (j > i) Then Exit While
If (k And 1) Then
p(i) += p(i - j)
Else
p(i) -= p(i - j)
End If
Wend
Next i
Return p(n)
End Function
Print !"\nPartitionsP: ";
For x As UInteger = 0 To 12
Print PartitionsP(x);" ";
Next x
Print !"\n\ndone"
Sleep

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Task/Partition-function-P/FreeBASIC/partition-function-p-2.basic Normal file
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' version 26-06-2021
' compile with: fbc -s console
#Include Once "gmp.bi"
Sub PartitionsP(max As ULong, p() As MpZ_ptr)
' based on Numericana code example
Dim As ULong a, b, i, k
Dim As Long j
Dim As Mpz_ptr s = Allocate(Len(__mpz_struct)) : Mpz_init(s)
Mpz_set_ui(p(0), 1)
For i = 1 To max
j = 1 : k = 1 : b = 2 : a = 5
While j > 0
' j = i - (3*k*k+k) \ 2
j = i - b : b = b + a : a = a + 3
If j >= 0 Then
If k And 1 Then Mpz_add(s, s, p(j)) Else Mpz_sub(s, s, p(j))
End If
j = j + k
If j >= 0 Then
If k And 1 Then Mpz_add(s, s, p(j)) Else Mpz_sub(s, s, p(j))
End If
k = k +1
Wend
Mpz_swap(p(i), s)
Next
Mpz_clear(s)
End Sub
' ------=< MAIN >=------
#Define max 6666
Dim As UInteger n
Dim As ZString Ptr ans
Dim As Double t = Timer
ReDim big_p(max) As Mpz_ptr
For n = 0 To max
big_p(n) = Allocate(Len(__mpz_struct)) : Mpz_init(big_p(n))
Next
PartitionsP(max, big_p())
ans = Mpz_get_str (0, 10, big_p(max))
Print "PartitionsP("; Str(max); ") = "; " "; *ans
For n = 0 To max
Mpz_clear(big_p(n))
Next
Print Using "time = ###.## ms"; (Timer - t) * 1000
' empty keyboard buffer
While InKey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End

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Task/Partition-function-P/Frink/partition-function-p.frink Normal file
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println[partitionCount[6666]]

59
Task/Partition-function-P/Go/partition-function-p.go Normal file
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package main
import (
"fmt"
"math/big"
"time"
)
var p []*big.Int
var pd []int
func partDiffDiff(n int) int {
if n&1 == 1 {
return (n + 1) / 2
}
return n + 1
}
func partDiff(n int) int {
if n < 2 {
return 1
}
pd[n] = pd[n-1] + partDiffDiff(n-1)
return pd[n]
}
func partitionsP(n int) {
if n < 2 {
return
}
psum := new(big.Int)
for i := 1; i <= n; i++ {
pdi := partDiff(i)
if pdi > n {
break
}
sign := int64(-1)
if (i-1)%4 < 2 {
sign = 1
}
t := new(big.Int).Mul(p[n-pdi], big.NewInt(sign))
psum.Add(psum, t)
}
p[n] = psum
}
func main() {
start := time.Now()
const N = 6666
p = make([]*big.Int, N+1)
pd = make([]int, N+1)
p[0], pd[0] = big.NewInt(1), 1
p[1], pd[1] = big.NewInt(1), 1
for n := 2; n <= N; n++ {
partitionsP(n)
}
fmt.Printf("p[%d)] = %d\n", N, p[N])
fmt.Printf("Took %s\n", time.Since(start))
}

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Task/Partition-function-P/Haskell/partition-function-p.hs Normal file
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{-# LANGUAGE DeriveFunctor #-}
------------------------------------------------------------
-- memoization utilities
data Memo a = Node a (Memo a) (Memo a)
deriving (Functor)
memo :: Integral a => Memo p -> a -> p
memo (Node a l r) n
| n == 0 = a
| odd n = memo l (n `div` 2)
| otherwise = memo r (n `div` 2 - 1)
nats :: Memo Int
nats =
Node
0
((+ 1) . (* 2) <$> nats)
((* 2) . (+ 1) <$> nats)
------------------------------------------------------------
-- calculating partitions
partitions :: Memo Integer
partitions = partitionP <$> nats
partitionP :: Int -> Integer
partitionP n
| n < 2 = 1
| otherwise = sum $ zipWith (*) signs terms
where
terms =
[ memo partitions (n - i)
| i <- takeWhile (<= n) ofsets
]
signs = cycle [1, 1, -1, -1]
ofsets :: [Int]
ofsets = scanl1 (+) $ mix [1, 3 ..] [1, 2 ..]
where
mix a b = concat $ zipWith (\x y -> [x, y]) a b
main :: IO ()
main = print $ partitionP 6666

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Task/Partition-function-P/J/partition-function-p.j Normal file
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pn =: -/@(+/)@:($:"0)@rec ` (x:@(0&=)) @. (0>:]) M.
rec=: - (-: (*"1) _1 1 +/ 3 * ]) @ (>:@i.@>.@%:@((2%3)&*))

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Task/Partition-function-P/Java/partition-function-p.java Normal file
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import java.math.BigInteger;
public class PartitionFunction {
public static void main(String[] args) {
long start = System.currentTimeMillis();
BigInteger result = partitions(6666);
long end = System.currentTimeMillis();
System.out.println("P(6666) = " + result);
System.out.printf("elapsed time: %d milliseconds\n", end - start);
}
private static BigInteger partitions(int n) {
BigInteger[] p = new BigInteger[n + 1];
p[0] = BigInteger.ONE;
for (int i = 1; i <= n; ++i) {
p[i] = BigInteger.ZERO;
for (int k = 1; ; ++k) {
int j = (k * (3 * k - 1))/2;
if (j > i)
break;
if ((k & 1) != 0)
p[i] = p[i].add(p[i - j]);
else
p[i] = p[i].subtract(p[i - j]);
j += k;
if (j > i)
break;
if ((k & 1) != 0)
p[i] = p[i].add(p[i - j]);
else
p[i] = p[i].subtract(p[i - j]);
}
}
return p[n];
}
}

18
Task/Partition-function-P/JavaScript/partition-function-p.js Normal file
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function p(n){
var a = new Array(n+1)
a[0] = 1n
for (let i = 1; i <= n; i++){
a[i] = 0n
for (let k = 1, s = 1; s <= i;){
a[i] += (k & 1 ? a[i-s]:-a[i-s])
k > 0 ? (s += k, k = -k):(k = -k+1, s = k*(3*k-1)/2)
}
}
return a[n]
}
var t = Date.now()
console.log("p(6666) = " + p(6666))
console.log("Computation time in ms: ", Date.now() - t)

22
Task/Partition-function-P/Jq/partition-function-p-1.jq Normal file
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def partitions($n):
def div2: (. - (.%2)) / 2;
reduce range(1; $n + 1) as $i ( {p: ([1] + [range(0;$n)|0])};
. + {k: 0, stop: false}
| until(.stop;
.k += 1
| (((.k * (3*.k - 1)) | div2) ) as $j
| if $j > $i then .stop=true
else if (.k % 2) == 1
then .p[$i] = .p[$i] + .p[$i - $j]
else .p[$i] = .p[$i] - .p[$i - $j]
end
| (((.k * (3*.k + 1)) | div2)) as $j
| if $j > $i then .stop=true
elif (.k % 2) == 1
then .p[$i] = .p[$i] + .p[$i - $j]
else .p[$i] = .p[$i] - .p[$i - $j]
end
end ))
| .p[$n] ;
[partitions(range(1;15))]

49
Task/Partition-function-P/Jq/partition-function-p-2.jq Normal file
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def partDiffDiff($n):
if ($n % 2) == 1 then ($n+1) / 2 else $n+1 end;
# in: {n, partDiffMemo}
# out: object with possibly updated memoization
def partDiff:
.n as $n
| if .partDiffMemo[$n] then .
elif $n<2 then .partDiffMemo[$n]=1
else ((.n=($n-1)) | partDiff)
| .partDiffMemo[$n] = .partDiffMemo[$n-1] + partDiffDiff($n-1)
end;
# in: {n, memo, partDiffMemo}
# where `.memo[i]` memoizes partitions(i)
# and `.partDiffMemo[i]` memoizes partDiff(i)
# out: object with possibly updated memoization
def partitionsM:
.n as $n
| if .memo[$n] then .
elif $n<2 then .memo[$n] = 1
else label $out
| foreach range(1; $n+2) as $i (.emit = false | .psum = 0;
if $i > $n then .emit = true
else ((.n = $i) | partDiff)
| .partDiffMemo[$i] as $pd
| if $pd > $n then .emit=true, break $out
else {psum, emit} as $local # for restoring relevant state
| ((.n = ($n-$pd)) | partitionsM)
| .memo[$n-$pd] as $increment
| . + $local # restore
| if (($i-1)%4)<2
then .psum += $increment
else .psum -= $increment
end
end
end;
select(.emit) )
| .memo[$n] = .psum
end ;
def partitionsP:
. as $n
| {n: $n, memo:[], partDiffMemo:[]}
| partitionsM
| .memo[$n];
# Stretch goal:
6666 | partitionsP

33
Task/Partition-function-P/Julia/partition-function-p.julia Normal file
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using Memoize
function partDiffDiff(n::Int)::Int
isodd(n) ? (n+1)÷2 : n+1
end
@memoize function partDiff(n::Int)::Int
n<2 ? 1 : partDiff(n-1)+partDiffDiff(n-1)
end
@memoize function partitionsP(n::Int)
T=BigInt
if n<2
one(T)
else
psum = zero(T)
for i ∈ 1:n
pd = partDiff(i)
if pd>n
break
end
if ((i-1)%4)<2
psum += partitionsP(n-pd)
else
psum -= partitionsP(n-pd)
end
end
psum
end
end
n=6666
@time println("p($n) = ", partitionsP(n))

28
Task/Partition-function-P/Lingo/partition-function-p.lingo Normal file
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-- returns number of partitions of n
on partitions(n, res_table)
if n < 2 then return 1
if voidP(res_table) then
res_table = []
res_table[n] = 0
else if res_table[n] then
return res_table[n]
end if
res = 0
i = 0
param = 1
repeat while param <= n
if i mod 4 < 2 then
res = res + partitions(n - param, res_table)
else
res = res - partitions(n - param, res_table)
end if
if i mod 2 then
param = param + i + 2
else
param = param + i / 2 + 1
end if
i = i + 1
end repeat
res_table[n] = res
return res
end

30
Task/Partition-function-P/Maple/partition-function-p.maple Normal file
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p:=proc(n)
option remember;
local k,s:=0,m;
for k from 1 while (m:=iquo(k*(3*k-1),2))<=n do
s-=(-1)^k*p(n-m);
od;
for k from 1 while (m:=iquo(k*(3*k+1),2))<=n do
s-=(-1)^k*p(n-m);
od;
s
end:
p(0):=1:
time(p(6666));
# 0.796
time(combinat[numbpart](6666));
# 0.406
p~([$1..20]);
# [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627]
combinat[numbpart]~([$1..20]);
# [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627]
p(1000)
# 24061467864032622473692149727991
combinat[numbpart](1000);
# 24061467864032622473692149727991

3
Task/Partition-function-P/Mathematica/partition-function-p.math Normal file
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PartitionsP /@ Range[15]
PartitionsP[666]
PartitionsP[6666]

27
Task/Partition-function-P/Nim/partition-function-p.nim Normal file
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import sequtils, strformat, times
import bignum
func partitions(n: int): Int =
var p = newSeqWith(n + 1, newInt())
p[0] = newInt(1)
for i in 1..n:
var k = 1
while true:
var j = k * (3 * k - 1) div 2
if j > i: break
if (k and 1) != 0:
inc p[i], p[i - j]
else:
dec p[i], p[i - j]
j = k * (3 * k + 1) div 2
if j > i: break
if (k and 1) != 0:
inc p[i], p[i - j]
else:
dec p[i], p[i - j]
inc k
result = p[n]
let t0 = cpuTime()
echo partitions(6666)
echo &"Elapsed time: {(cpuTime() - t0) * 1000:.2f} ms"

29
Task/Partition-function-P/Perl/partition-function-p.pl Normal file
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use strict;
use warnings;
no warnings qw(recursion);
use Math::AnyNum qw(:overload);
use Memoize;
memoize('partitionsP');
memoize('partDiff');
sub partDiffDiff { my($n) = @_; $n%2 != 0 ? ($n+1)/2 : $n+1 }
sub partDiff { my($n) = @_; $n<2 ? 1 : partDiff($n-1) + partDiffDiff($n-1) }
sub partitionsP {
my($n) = @_;
return 1 if $n < 2;
my $psum = 0;
for my $i (1..$n) {
my $pd = partDiff($i);
last if $pd > $n;
if ( (($i-1)%4) < 2 ) { $psum += partitionsP($n-$pd) }
else { $psum -= partitionsP($n-$pd) }
}
return $psum
}
print partitionsP($_) . ' ' for 0..25; print "\n";
print partitionsP(6666) . "\n";

42
Task/Partition-function-P/Phix/partition-function-p.phix Normal file
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(phixonline)-->
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">partDiffDiff</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)/(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">+</span><span style="color: #7060A8;">and_bits</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">))</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">pd</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">}</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">partDiff</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">while</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">></span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pd</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">pd</span> <span style="color: #0000FF;">&=</span> <span style="color: #000000;">pd</span><span style="color: #0000FF;">[$]</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">partDiffDiff</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pd</span><span style="color: #0000FF;">))</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">pd</span><span style="color: #0000FF;">[</span><span style="color: #7060A8;">max</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)]</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">include</span> <span style="color: #004080;">mpfr</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">pn</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)}</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">partitionsP</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">mpz</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">while</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">></span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pn</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">nn</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pn</span><span style="color: #0000FF;">)+</span><span style="color: #000000;">1</span>
<span style="color: #004080;">mpz</span> <span style="color: #000000;">psum</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">nn</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">pd</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">partDiff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">pd</span><span style="color: #0000FF;">></span><span style="color: #000000;">nn</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">sgn</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">remainder</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">4</span><span style="color: #0000FF;">)<</span><span style="color: #000000;">2</span> <span style="color: #0000FF;">?</span> <span style="color: #000000;">1</span> <span style="color: #0000FF;">:</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">mpz</span> <span style="color: #000000;">pnmpd</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">pn</span><span style="color: #0000FF;">[</span><span style="color: #7060A8;">max</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">nn</span><span style="color: #0000FF;">-</span><span style="color: #000000;">pd</span><span style="color: #0000FF;">)]</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">sgn</span><span style="color: #0000FF;">=-</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
<span style="color: #7060A8;">mpz_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">psum</span><span style="color: #0000FF;">,</span><span style="color: #000000;">psum</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pnmpd</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">else</span>
<span style="color: #7060A8;">mpz_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">psum</span><span style="color: #0000FF;">,</span><span style="color: #000000;">psum</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pnmpd</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #000000;">pn</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">append</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pn</span><span style="color: #0000FF;">,</span><span style="color: #000000;">psum</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">pn</span><span style="color: #0000FF;">[</span><span style="color: #7060A8;">max</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)]</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">t0</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">=</span><span style="color: #000000;">6666</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"p(%d) = %s (%s)\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">partitionsP</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)),</span><span style="color: #7060A8;">elapsed</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">time</span><span style="color: #0000FF;">()-</span><span style="color: #000000;">t0</span><span style="color: #0000FF;">)})</span>
<!--

26
Task/Partition-function-P/Picat/partition-function-p.picat Normal file
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/* Picat 3.0#5 */
/* Author: Hakan Kjellerstrand */
table
partition1(0) = 1.
partition1(N) = P =>
S = 0,
K = 1,
M = (K*(3*K-1)) // 2,
while (M <= N)
S := S - ((-1)**K)*partition1(N-M),
K := K + 1,
M := (K*(3*K-1)) // 2
end,
K := 1,
M := (K*(3*K+1)) // 2,
while (M <= N)
S := S - ((-1)**K)*partition1(N-M),
K := K + 1,
M := (K*(3*K+1)) // 2
end,
P = S.
Picat> time(println('p(6666)'=partition1(6666)))
p(6666) = 193655306161707661080005073394486091998480950338405932486880600467114423441282418165863
CPU time 0.206 seconds.

18
Task/Partition-function-P/PicoLisp/partition-function-p.l Normal file
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(de gpentagonals (Max)
(make
(let (N 0 M 1)
(loop
(inc 'N (if (=0 (& M 1)) (>> 1 M) M))
(T (> N Max))
(link N)
(inc 'M)))))
(de p (N)
(cache '(NIL) N
(if (=0 N)
1
(let (Sum 0 Sgn 0)
(for G (gpentagonals N)
((if (< Sgn 2) 'inc 'dec) 'Sum (p (- N G)))
(setq Sgn (& 3 (inc Sgn))))
Sum))))

15
Task/Partition-function-P/Prolog/partition-function-p.pro Normal file
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/* SWI-Prolog 8.3.21 */
/* Author: Jan Burse */
:- table p/2.
p(0, 1) :- !.
p(N, X) :-
aggregate_all(sum(Z), (between(1,inf,K), M is K*(3*K-1)//2,
(M>N, !, fail; L is N-M, p(L,Y), Z is (-1)^K*Y)), A),
aggregate_all(sum(Z), (between(1,inf,K), M is K*(3*K+1)//2,
(M>N, !, fail; L is N-M, p(L,Y), Z is (-1)^K*Y)), B),
X is -A-B.
?- time(p(6666,X)).
% 13,962,294 inferences, 2.610 CPU in 2.743 seconds (95% CPU, 5350059 Lips)
X = 1936553061617076610800050733944860919984809503384
05932486880600467114423441282418165863.

51
Task/Partition-function-P/Python/partition-function-p-1.py Normal file
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from itertools import islice
def posd():
"diff between position numbers. 1, 2, 3... interleaved with 3, 5, 7..."
count, odd = 1, 3
while True:
yield count
yield odd
count, odd = count + 1, odd + 2
def pos_gen():
"position numbers. 1 3 2 5 7 4 9 ..."
val = 1
diff = posd()
while True:
yield val
val += next(diff)
def plus_minus():
"yield (list_offset, sign) or zero for Partition calc"
n, sign = 0, [1, 1]
p_gen = pos_gen()
out_on = next(p_gen)
while True:
n += 1
if n == out_on:
next_sign = sign.pop(0)
if not sign:
sign = [-next_sign] * 2
yield -n, next_sign
out_on = next(p_gen)
else:
yield 0
def part(n):
"Partition numbers"
p = [1]
p_m = plus_minus()
mods = []
for _ in range(n):
next_plus_minus = next(p_m)
if next_plus_minus:
mods.append(next_plus_minus)
p.append(sum(p[offset] * sign for offset, sign in mods))
return p[-1]
print("(Intermediaries):")
print(" posd:", list(islice(posd(), 10)))
print(" pos_gen:", list(islice(pos_gen(), 10)))
print(" plus_minus:", list(islice(plus_minus(), 15)))
print("\nPartitions:", [part(x) for x in range(15)])

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Task/Partition-function-P/Python/partition-function-p-2.py Normal file
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def par_primes():
"Prime number generator from the partition machine"
p = [1]
p_m = plus_minus()
mods = []
n = 0
while True:
n += 1
next_plus_minus = next(p_m)
if next_plus_minus:
mods.append(next_plus_minus)
p.append(sum(p[offset] * sign for offset, sign in mods))
if p[0] + 1 == p[-1]:
yield p[0]
p[0] += 1
yield p
print("\nPrimes:", list(islice(par_primes(), 15)))

30
Task/Partition-function-P/Python/partition-function-p-3.py Normal file
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from typing import List
def partitions(n: int) -> int:
"""Count partitions."""
p: List[int] = [1] + [0] * n
for i in range(1, n + 1):
k: int = 0
while True:
k += 1
j: int = (k * (3*k - 1)) // 2
if (j > i):
break
if (k & 1):
p[i] += p[i - j]
else:
p[i] -= p[i - j]
j = (k * (3*k + 1)) // 2
if (j > i):
break
if (k & 1):
p[i] += p[i - j]
else:
p[i] -= p[i - j]
return p[n]
if __name__ == '__main__':
print("\nPartitions:", [partitions(x) for x in range(15)])

9
Task/Partition-function-P/Quackery/partition-function-p.quackery Normal file
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[ 1 swap
dup 0 = iff drop done
[ 2dup = iff [ 2drop 1 ] done
2dup > iff [ 2drop 0 ] done
2dup dip 1+ recurse
unrot over - recurse + ] ] is partitions ( n --> n )
say "Partitions of 0 to 29" cr
30 times [ i^ partitions echo sp ]

31
Task/Partition-function-P/REXX/partition-function-p-1.rexx Normal file
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/*REXX program calculates and displays a specific value (or a range of) partitionsP(N).*/
numeric digits 1000 /*able to handle some ginormous numbers*/
parse arg lo hi . /*obtain optional arguments from the CL*/
if lo=='' | lo=="," then lo= 0 /*Not specified? Then use the default.*/
if hi=='' | hi=="," then hi= lo /* " " " " " " */
@.= 0; @.0= 1; @.1= 1; @.2= 2; @.3= 3; @.4= 5 /*assign default value and low values. */
!.= @.; !.1= 1; !.3= 1; !.5= 1; !.7= 1; !.9= 1 /*assign default value and even digits.*/
w= length( commas(hi) ) /*W: is used for aligning the index. */
do j=lo to hi /*compute a range of partitionsP. */
say right( commas(j), w) ' ' commas( partP(j) )
end /*j*/
exit 0 /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: parse arg ?; do jc=length(?)-3 to 1 by -3; ?=insert(',', ?, jc); end; return ?
/*──────────────────────────────────────────────────────────────────────────────────────*/
partP: procedure expose @. !.; parse arg n /*obtain number (index) for computation*/
if @.n\==0 then return @.n /*Is it already computed? Return it. */
#= 0 /*initialize part P number.*/
do k=1 for n; z= n - (k*3 - 1) * k % 2 /*compute the partition P num*/
if z<0 then leave /*Is Z negative? Then leave.*/
if @.z==0 then x= partP(z) /*use recursion if not known.*/
else x= @.z /*use the pre─computed number*/
z= z - k /*subtract index (K) from Z. */
if z<0 then y= 0 /*Is Z negative? Then set Y=0*/
else if @.z==0 then y= partP(z) /*use recursion if not known.*/
else y= @.z /*use the pre─computed number*/
if k//2 then #= # + x + y /*Odd? Then sum X and Y.*/
else #= # - (x + y) /*Even? " subtract " " " */
end /*k*/
@.n= #; return # /*define and return partitionsP of N. */

32
Task/Partition-function-P/REXX/partition-function-p-2.rexx Normal file
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/*REXX program calculates and displays a specific value (or a range of) partitionsP(N).*/
numeric digits 1000 /*able to handle some ginormous numbers*/
parse arg lo hi . /*obtain optional arguments from the CL*/
if lo=='' | lo=="," then lo= 0 /*Not specified? Then use the default.*/
if hi=='' | hi=="," then hi= lo /* " " " " " " */
@.= 0; @.0= 1; @.1= 1; @.2= 2; @.3= 3; @.4= 5 /*default values for some low numbers. */
!.= @.; !.1= 1; !.3= 1; !.5= 1; !.7= 1; !.9= 1 /* " " " all the 1─digit #s*/
w= length( commas(hi) ) /*W: is used for aligning the index. */
do j=lo to hi /*compute a range of partitionsP. */
say right( commas(j), w) ' ' commas( partP(j) )
end /*j*/
exit 0 /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: parse arg ?; do jc=length(?)-3 to 1 by -3; ?=insert(',', ?, jc); end; return ?
/*──────────────────────────────────────────────────────────────────────────────────────*/
partP: procedure expose @. !.; parse arg n /*obtain number (index) for computation*/
if @.n\==0 then return @.n /*Is it already computed? Return it. */
#= 0 /*initialize part P number.*/
do k=1 for n; z= n - (k+k+k - 1) * k % 2 /*compute the partition P num*/
if z<0 then leave /*Is Z negative? Then leave.*/
if @.z==0 then x= partP(z) /*use recursion if not known.*/
else x= @.z /*use the pre─computed number*/
z= z - k /*subtract index (K) from Z. */
if z<0 then y= 0 /*Is Z negative? Then set Y=0*/
else if @.z==0 then y= partP(z) /*use recursion if not known.*/
else y= @.z /*use the pre─computed number*/
parse var k '' -1 _ /*obtain K's last decimal dig*/
if !._ then #= # + x + y /*Odd? Then sum X and Y.*/
else #= # - (x + y) /*Even? " subtract " " " */
end /*k*/
@.n= #; return # /*define and return partitionsP of N. */

34
Task/Partition-function-P/REXX/partition-function-p-3.rexx Normal file
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/*REXX program calculates and displays a specific value (or a range of) partitionsP(N).*/
numeric digits 1000 /*able to handle some ginormous numbers*/
parse arg lo hi . /*obtain optional arguments from the CL*/
if lo=='' | lo=="," then lo= 0 /*Not specified? Then use the default.*/
if hi=='' | hi=="," then hi= lo /* " " " " " " */
@.= 0; @.0= 1; @.1= 1; @.2= 2; @.3= 3; @.4= 5 /*default values for some low numbers. */
!.= @.; !.1= 1; !.3= 1; !.5= 1; !.7= 1; !.9= 1 /* " " " all the 1─digit #s*/
w= length( commas(hi) ) /*W: is used for aligning the index. */
do i=1 for hi; a.i= (i+i+i - 1) * i % 2 /*calculate HI expressions (for partP).*/
end /*i*/
do j=lo to hi /*compute a range of partitionsP. */
say right( commas(j), w) ' ' commas( partP(j) )
end /*j*/
exit 0 /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: parse arg ?; do jc=length(?)-3 to 1 by -3; ?=insert(',', ?, jc); end; return ?
/*──────────────────────────────────────────────────────────────────────────────────────*/
partP: procedure expose @. !. a.; parse arg n /*obtain number (index) for computation*/
if @.n\==0 then return @.n /*Is it already computed? Return it. */
#= 0 /*initialize part P number.*/
do k=1 for n; z= n - a.k /*compute the partition P num*/
if z<0 then leave /*Is Z negative? Then leave.*/
if @.z==0 then x= partP(z) /*use recursion if not known.*/
else x= @.z /*use the pre─computed number*/
z= z - k /*subtract index (K) from Z. */
if z<0 then y= 0 /*Is Z negative? Then set Y=0*/
else if @.z==0 then y= partP(z) /*use recursion if not known.*/
else y= @.z /*use the pre─computed number*/
parse var k '' -1 _ /*obtain K's last decimal dig*/
if !._ then #= # + x + y /*Odd? Then sum X and Y.*/
else #= # - (x + y) /*Even? " subtract " " " */
end /*k*/
@.n= #; return # /*define and return partitionsP of N. */

28
Task/Partition-function-P/Racket/partition-function-p.rkt Normal file
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#lang racket
(require math/number-theory)
(define σ
(let ((memo (make-hash)))
(λ (z)
(hash-ref! memo z
(λ () (apply + (divisors z)))))))
(define p
(let ((memo (make-hash '((0 . 1)))))
(λ (n)
(hash-ref!
memo n
(λ ()
(let ((r (if (zero? n) 1
(/ (for/sum ((k (in-range (sub1 n) -1 -1)))
(* (σ (- n k))
(p k)))
n))))
(when (zero? (modulo n 1000)) (displayln (cons n r) (current-error-port)))
r))))))
(map p (range 1 30))
(p 666)
(p 1000)
(p 10000)

6
Task/Partition-function-P/Raku/partition-function-p.raku Normal file
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my @P = 1, { p(++$) } … *;
my @i = lazy [\+] flat 1, ( 1..* Z (1..*).map: * × 2 + 1 );
sub p ($n) { sum @P[$n X- @i] Z× (flat (1, 1, -1, -1) xx *) }
put @P[^26];
put @P[6666];

46
Task/Partition-function-P/Rust/partition-function-p.rust Normal file
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// [dependencies]
// rug = "1.11"
use rug::Integer;
fn partitions(n: usize) -> Integer {
let mut p = Vec::with_capacity(n + 1);
p.push(Integer::from(1));
for i in 1..=n {
let mut num = Integer::from(0);
let mut k = 1;
loop {
let mut j = (k * (3 * k - 1)) / 2;
if j > i {
break;
}
if (k & 1) == 1 {
num += &p[i - j];
} else {
num -= &p[i - j];
}
j += k;
if j > i {
break;
}
if (k & 1) == 1 {
num += &p[i - j];
} else {
num -= &p[i - j];
}
k += 1;
}
p.push(num);
}
p[n].clone()
}
fn main() {
use std::time::Instant;
let n = 6666;
let now = Instant::now();
let result = partitions(n);
let time = now.elapsed();
println!("P({}) = {}", n, result);
println!("elapsed time: {} microseconds", time.as_micros());
}

1
Task/Partition-function-P/Sidef/partition-function-p-1.sidef Normal file
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say partitions(6666) # very fast

23
Task/Partition-function-P/Sidef/partition-function-p-2.sidef Normal file
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func partitionsP(n) {
func (n) is cached {
n <= 1 && return n
var a = sum(1..floor((sqrt(24*n + 1) + 1)/6), {|k|
(-1)**(k-1) * __FUNC__(n - ((k*(3*k - 1)) >> 1))
})
var b = sum(1..ceil((sqrt(24*n + 1) - 7)/6), {|k|
(-1)**(k-1) * __FUNC__(n - ((k*(3*k + 1)) >> 1))
})
a + b
}(n+1)
}
var t = Time.micro
say partitionsP.map(0..25).join(' ')
say partitionsP(6666)
say ("Took %.4f seconds" % Time.micro-t)

44
Task/Partition-function-P/Swift/partition-function-p.swift Normal file
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import BigInt
func partitions(n: Int) -> BigInt {
var p = [BigInt(1)]
for i in 1...n {
var num = BigInt(0)
var k = 1
while true {
var j = (k * (3 * k - 1)) / 2
if j > i {
break
}
if k & 1 == 1 {
num += p[i - j]
} else {
num -= p[i - j]
}
j += k
if j > i {
break
}
if k & 1 == 1 {
num += p[i - j]
} else {
num -= p[i - j]
}
k += 1
}
p.append(num)
}
return p[n]
}
print("partitions(6666) = \(partitions(n: 6666))")

36
Task/Partition-function-P/Wren/partition-function-p.wren Normal file
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import "/big" for BigInt
var p = []
var pd = []
var partDiffDiff = Fn.new { |n| (n&1 == 1) ? (n + 1)/2 : n + 1 }
var partDiff = Fn.new { |n|
if (n < 2) return 1
pd[n] = pd[n-1] + partDiffDiff.call(n-1)
return pd[n]
}
var partitionsP = Fn.new { |n|
if (n < 2) return
var psum = BigInt.zero
for (i in 1..n) {
var pdi = partDiff.call(i)
if (pdi > n) break
var sign = (i-1)%4 < 2 ? 1 : -1
psum = psum + p[n-pdi] * sign
}
p[n] = psum
}
var start = System.clock
var N = 6666
p = List.filled(N+1, null)
pd = List.filled(N+1, 0)
p[0] = BigInt.one
p[1] = BigInt.one
pd[0] = 1
pd[1] = 1
for (n in 2..N) partitionsP.call(n)
System.print("p[%(N)] = %(p[N])")
System.print("Took %(System.clock - start) seconds")