A384928 Number of 2-dense sublists of divisors of the n-th triangular number.

1, 1, 2, 1, 2, 3, 4, 1, 1, 3, 4, 1, 2, 3, 4, 1, 2, 5, 6, 3, 1, 5, 4, 1, 1, 5, 6, 1, 4, 5, 6, 1, 1, 5, 6, 1, 2, 3, 6, 1, 2, 7, 8, 3, 1, 3, 4, 1, 1, 5, 6, 3, 4, 7, 3, 1, 1, 5, 4, 1, 2, 3, 8, 1, 1, 7, 8, 3, 3, 5, 6, 1, 2, 3, 6, 1, 4, 5, 8, 1, 1, 7, 4, 1, 1, 7, 6, 1, 4, 5, 3, 3, 3, 5, 8, 1, 2, 5, 5, 1, 6

Offset

0,3

Comments

By definition a(n) is also the number of 2-dense sublists of divisors of the n-th generalized hexagonal number.

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 odd indexed terms are odd.

For odd n = 2k + 1 the n-th triangular number t(n) = (2k+1)*(k+1) satisfies A003056(t(n)) = 2k + 1; i.e., n is an odd divisor of t(n). Therefore SRS(t(n)) at the diagonal is positive and SRS(t(n)) has an odd number of parts. Since the number of parts equals the number of 2-dense sublists (see link in A384149), the conjecture is true. - Hartmut F. W. Hoft, Mar 25 2026

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Formula

a(n) = A237271(A000217(n)) for n >= 1 (conjectured).

Example

For n = 5 the 5th triangular 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(5) = 3.
For n = 12 the 12th triangular number is 78. The list of divisors of 78 is [1, 2, 3, 6, 13, 26, 39, 78]. There are two 2-dense sublists of divisors of 78, they are [1, 2, 3, 6] and [13, 26, 39, 78], so a(12) = 2. Note that 78 is also the first practical number A005153 not in the sequence of the 2-dense numbers A174973.

Mathematica

A384928[n_] := Length[Split[Divisors[PolygonalNumber[n]], #2 <= 2*# &]];
Array[A384928, 100, 0] (* Paolo Xausa, Aug 14 2025 *)

Crossrefs

Cf. A000217, A005153, A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A384930, A384931, A386984 (a bisection), A386989.

Sequence in context: A316231 A014781 A214500 * A066016 A098068 A194070

Adjacent sequences: A384925 A384926 A384927 * A384929 A384930 A384931

Keyword

nonn

Author

Omar E. Pol, Aug 08 2025

Status

approved