A057568 Number of partitions of n where n divides the product of the parts.
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, 809, 726, 1, 1787, 1, 4261, 735, 260, 1925, 9567, 1, 437, 1877, 16402, 1, 14630, 1, 9861, 33057, 1152, 1, 102082, 19393, 57330, 10159, 30706, 1, 207706, 47927, 200652
Offset: 1
Keywords
Examples
From _Gus Wiseman_, Jul 04 2019: (Start)
The a(1) = 1 through a(9) = 5 partitions are the following. The Heinz numbers of these partitions are given by A326149.
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(22) (321) (44) (63)
(422) (333)
(2222) (3321)
(4211) (33111)
(22211)
(End)
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..1000 (terms n=1..73 from Antti Karttunen)
Crossrefs
Programs
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Maple
b:= proc(n, i, t) option remember; `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/igcd(i, t))))) end: a:= n-> `if`(isprime(n), 1, b(n$3)): seq(a(n), n=1..70); # Alois P. Heinz, Dec 20 2017 -
Mathematica
Table[Length[Select[IntegerPartitions[n],Divisible[Times@@#,n]&]],{n,20}] (* Gus Wiseman, Jul 04 2019 *) 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]]]]]; a[n_] := If[PrimeQ[n], 1, b[n, n, n]]; Array[a, 70] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *) -
Scheme
;; This is a naive algorithm that scans over all partitions of each n. For fold_over_partitions_of see A000793. (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))) ;; Antti Karttunen, Dec 20 2017
Extensions
More terms from James Sellers, Oct 09 2000