A332887 a(n) is the number of partition numbers > 1 that are proper divisors of the n-th partition number.

0, 0, 0, 0, 0, 2, 2, 4, 3, 2, 2, 0, 3, 3, 4, 2, 4, 3, 2, 4, 2, 1, 4, 3, 4, 3, 3, 2, 2, 3, 2, 2, 2, 2, 0, 3, 2, 10, 4, 5, 3, 3, 1, 1, 2, 2, 2, 2, 3, 2, 5, 2, 5, 2, 1, 2, 4, 4, 0, 8, 3, 1, 4, 1, 2, 0, 4, 1, 2, 3, 1, 0, 4, 9, 1, 0, 3, 2, 2, 1, 5, 3, 4, 1, 1, 1

Offset

2,6

Comments

Conjecture: every nonnegative integer occurs infinitely many times.

Links

Table of n, a(n) for n=2..87.

Formula

a(n) = A332886(n) - 2 for n >= 2.

Example

Let p(n) = A000041(n) = number of partitions of n.  Then p(9) = 30, which is divisible by these 6 partition numbers: p(1) = 1, p(2) = 2, p(3) = 3, p(4) = 5, p(7) = 15, and p(9) = 30; discounting p(1) and p(9) leaves a(9) = 4 proper divisors.

Mathematica

p[n_] := PartitionsP[n]; t[n_] := Table[p[k], {k, 0, n}]
u = -2 + Table[Length[Intersection[t[n], Divisors[p[n]]]], {n, 2, 130}]

Prog

(PARI) a(n) = my(nbp=numbpart(n)); sum(k=2, n-1, (nbp % numbpart(k)) == 0); \\ Michel Marcus, Feb 29 2020

Crossrefs

Cf. A000041 (partition numbers), A332886.

Sequence in context: A105114 A166284 A098086 * A306323 A317015 A341902

Adjacent sequences: A332884 A332885 A332886 * A332888 A332889 A332890

Keyword

nonn

Author

Clark Kimberling, Feb 29 2020

Status

approved