A332886 a(n) is the number of partition numbers that divide the n-th partition number (counted without multiplicity).

1, 1, 2, 2, 2, 2, 2, 4, 4, 6, 5, 4, 4, 2, 5, 5, 6, 4, 6, 5, 4, 6, 4, 3, 6, 5, 6, 5, 5, 4, 4, 5, 4, 4, 4, 4, 2, 5, 4, 12, 6, 7, 5, 5, 3, 3, 4, 4, 4, 4, 5, 4, 7, 4, 7, 4, 3, 4, 6, 6, 2, 10, 5, 3, 6, 3, 4, 2, 6, 3, 4, 5, 3, 2, 6, 11, 3, 2, 5, 4, 4, 3, 7, 5, 6

Offset

0,3

Comments

Conjecture: every positive integer occurs infinitely many times.

Links

Table of n, a(n) for n=0..84.

Formula

a(n) = A332887(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; thus a(9) = 6.
a(1) = 1 counts p(0) = 1 and p(1) = 1 as 1 divisor, as opposed to 2 numbers that have 1 as a divisor.

Mathematica

p[n_] := PartitionsP[n]; t[n_] := Table[p[k], {k, 1, n}]
Table[Length[Intersection[t[n], Divisors[p[n]]]], {n, 1, 130}]

Prog

(PARI) a(n) = if (n==0, 1, my(nbp=numbpart(n)); sum(k=1, n, (nbp % numbpart(k)) == 0)); \\ Michel Marcus, Feb 29 2020; corrected Jun 15 2022

Crossrefs

Cf. A000041 (partition numbers), A332887.

Sequence in context: A010336 A054537 A029104 * A085543 A294226 A083499

Adjacent sequences: A332883 A332884 A332885 * A332887 A332888 A332889

Keyword

nonn

Author

Clark Kimberling, Feb 29 2020

Status

approved