A236103 Number of distinct partition numbers dividing n.

1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 1, 3, 4, 2, 1, 3, 1, 3, 3, 4, 1, 3, 2, 2, 2, 3, 1, 6, 1, 2, 3, 2, 3, 3, 1, 2, 2, 3, 1, 5, 1, 4, 4, 2, 1, 3, 2, 3, 2, 2, 1, 3, 3, 4, 2, 2, 1, 6, 1, 2, 3, 2, 2, 5, 1, 2, 2, 4, 1, 3, 1, 2, 4, 2, 4, 3, 1, 3, 2, 2, 1, 5, 2, 2, 2, 4, 1, 6

Offset: 1

Author

Omar E. Pol, Jan 21 2014

Keywords

• nonn

Examples

			For n = 20 the divisors of 20 are 1, 2, 4, 5, 10, 20 and three of them are also partition numbers: 1, 2, 5, so a(20) = 3.
For n = 42 the divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42 and five of them are also partition numbers: 1, 2, 3, 7, 42, so a(42) = 5.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000041, A001221, A049575, A078506, A085543, A167392, A236102, A236105, A236107, A236108.

Programs

Mathematica

p = {1}; Table[If[n >= Last@p, AppendTo[p, PartitionsP[1 + Length@p]]]; Length@Select[p, Mod[n, #] == 0 &], {n, 90}] (* Giovanni Resta, Jan 22 2014 *)

Formula

From Amiram Eldar, Jan 01 2024: (Start)

a(n) = Sum_{d|n} A167392(d).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A078506 = 2.510597... . (End)