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A057568 Number of partitions of n where n divides the product of the parts.

%I A057568 #31 Jul 02 2025 16:02:00
%S A057568 1,1,1,2,1,2,1,6,5,5,1,22,1,11,23,80,1,113,1,150,85,45,1,737,226,84,
%T A057568 809,726,1,1787,1,4261,735,260,1925,9567,1,437,1877,16402,1,14630,1,
%U A057568 9861,33057,1152,1,102082,19393,57330,10159,30706,1,207706,47927,200652
%N A057568 Number of partitions of n where n divides the product of the parts.
%H A057568 Alois P. Heinz, <a href="/A057568/b057568.txt">Table of n, a(n) for n = 1..1000</a> (terms n=1..73 from Antti Karttunen)
%e A057568 From _Gus Wiseman_, Jul 04 2019: (Start)
%e A057568 The a(1) = 1 through a(9) = 5 partitions are the following. The Heinz numbers of these partitions are given by A326149.
%e A057568   (1)  (2)  (3)  (4)   (5)  (6)    (7)  (8)      (9)
%e A057568                  (22)       (321)       (44)     (63)
%e A057568                                         (422)    (333)
%e A057568                                         (2222)   (3321)
%e A057568                                         (4211)   (33111)
%e A057568                                         (22211)
%e A057568 (End)
%p A057568 b:= proc(n, i, t) option remember; `if`(n=0,
%p A057568       `if`(t=1, 1, 0), `if`(i<1, 0, b(n, i-1, t)+
%p A057568       `if`(i>n, 0, b(n-i, min(i, n-i), t/igcd(i, t)))))
%p A057568     end:
%p A057568 a:= n-> `if`(isprime(n), 1, b(n$3)):
%p A057568 seq(a(n), n=1..70);  # _Alois P. Heinz_, Dec 20 2017
%t A057568 Table[Length[Select[IntegerPartitions[n],Divisible[Times@@#,n]&]],{n,20}] (* _Gus Wiseman_, Jul 04 2019 *)
%t A057568 b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 1, 1, 0], If[i < 1, 0, b[n, i - 1, t] + If[i > n, 0, b[n - i, Min[i, n - i], t/GCD[i, t]]]]];
%t A057568 a[n_] := If[PrimeQ[n], 1, b[n, n, n]];
%t A057568 Array[a, 70] (* _Jean-François Alcover_, May 21 2021, after _Alois P. Heinz_ *)
%o A057568 (Scheme)
%o A057568 ;; This is a naive algorithm that scans over all partitions of each n. For fold_over_partitions_of see A000793.
%o A057568 (define (A057568 n) (let ((z (list 0))) (fold_over_partitions_of n 1 * (lambda (partprod) (if (zero? (modulo partprod n)) (set-car! z (+ 1 (car z)))))) (car z)))
%o A057568 ;; _Antti Karttunen_, Dec 20 2017
%Y A057568 Cf. A028422, A057567, A096276, A113309, A114324, A318950, A319000, A319005, A326149, A326152.
%K A057568 nonn
%O A057568 1,4
%A A057568 _Leroy Quet_, Oct 04 2000
%E A057568 More terms from _James Sellers_, Oct 09 2000