A386984 Number of 2-dense sublists of divisors of the n-th hexagonal number.

1, 1, 1, 3, 1, 3, 1, 3, 1, 5, 3, 5, 1, 5, 1, 5, 1, 5, 1, 3, 1, 7, 3, 3, 1, 5, 3, 7, 1, 5, 1, 3, 1, 7, 3, 5, 1, 3, 1, 5, 1, 7, 1, 7, 1, 5, 3, 5, 1, 5, 1, 7, 3, 5, 1, 7, 1, 7, 5, 7, 1, 5, 3, 3, 1, 7, 1, 7, 1, 7, 5, 5, 1, 7, 1, 7, 3, 5, 1, 3, 1, 9, 3, 7, 1, 7, 1, 7, 1, 5, 1, 5, 1, 9, 3, 3, 1, 3, 1, 9, 1

Offset: 0

Author

Omar E. Pol, Aug 11 2025

Keywords

• nonn
• changed

Comments

In a 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.

Conjecture: all terms are odd.

Examples

			For n = 3 the third positive hexagonal number is 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(3) = 3.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000

Crossrefs

Bisection of A384928.

Cf. A000384, A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A384930, A384931, A386989.

Programs

Mathematica

A386984[n_] := Length[Split[Divisors[PolygonalNumber[6, n]], #2 <= 2*# &]];
Array[A386984, 100, 0] (* Paolo Xausa, Aug 29 2025 *)

Formula

a(n) = A237271(A000384(n)) for n >= 1 (conjectured).