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

0,4

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.

Links

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

Formula

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

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A270559 A231727 A368876 * A382447 A270616 A394027

Adjacent sequences: A384928 A384929 A384930 * A384932 A384933 A384934

Keyword

nonn

Author

Omar E. Pol, Jul 30 2025

Extensions

More terms from Alois P. Heinz, Jul 30 2025

Status

approved