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

0,4

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.

Since the hexagonal numbers are the odd-numbered triangular numbers, the proof in A384928 proves this conjecture. - Hartmut F. W. Hoft, Mar 25 2026

Links

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

Formula

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

Example

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.

Mathematica

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

Crossrefs

Bisection of A384928.

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

Sequence in context: A294787 A294788 A254578 * A348610 A296120 A050345

Adjacent sequences: A386981 A386982 A386983 * A386985 A386986 A386987

Keyword

nonn

Author

Omar E. Pol, Aug 11 2025

Status

approved