A384931 Number of 2-dense sublists of divisors of the number of partitions of n.

1, 1, 1, 2, 2, 2, 2, 3, 2, 1, 1, 1, 3, 2, 3, 1, 5, 6, 4, 4, 5, 1, 2, 4, 3, 4, 1, 5, 4, 7, 2, 4, 9, 10, 4, 9, 2, 6, 9, 3, 1, 9, 4, 11, 8, 4, 3, 3, 8, 12, 4, 11, 7, 10, 5, 3, 7, 2, 2, 1, 8, 5, 6, 8, 5, 2, 1, 3, 10, 6, 1, 6, 8, 7, 1, 1, 4, 2, 7, 9, 3, 4, 9, 6, 2

Offset: 0

Author

Omar E. Pol, Jul 30 2025

Keywords

• nonn
• changed

Comments

In a 2-dense sublist of divisors of k the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of k.

The 2-dense sublists of divisors of k are the maximal sublists whose terms increase by a factor of at most 2.

Examples

			For n = 7 the number of partitions of 7 is A000041(7) = 15. The list of divisors of 15 is [1, 3, 5, 15]. There are three 2-dense sublists of divisors of 15, they are [1], [3, 5], [15], so a(7) = 3.
For n = 19 the number of partitions of 19 is A000041(19) = 490. The list of divisors of 490 is [1, 2, 5, 7, 10, 14, 35, 49, 70, 98, 245, 490]. There are four 2-dense sublists of divisors of 490, they are [1, 2], [5, 7, 10, 14], [35, 49, 70, 98], [245, 490], so a(19) = 4.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..4000

Crossrefs

Cf. A000041, A174973 (2-dense numbers), A237271, A384149, A384222, A384225, A384226, A384930.

Programs

Mathematica

A384931[n_] := Length[Split[Divisors[PartitionsP[n]], #2 <= 2*# &]];
Array[A384931, 100, 0] (* Paolo Xausa, Aug 28 2025 *)

Formula

a(n) = A237271(A000041(n)). Conjectured.

Extensions

More terms from Alois P. Heinz, Jul 30 2025