A384230 Number of subparts in the central part of the symmetric representation of sigma(n).

1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 0, 2, 1, 0, 3, 0, 2, 0, 0, 0, 2, 1, 0, 0, 2, 0, 4, 0, 1, 0, 0, 2, 3, 0, 0, 0, 2, 0, 4, 0, 0, 4, 0, 0, 2, 1, 1, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 0, 2, 1, 0, 4, 0, 0, 0, 2, 0, 3, 0, 0, 0, 0, 2, 0, 0, 2, 1, 0, 0, 4, 0, 0, 0

Offset: 1

Author

Omar E. Pol, Jun 29 2025

Keywords

• nonn
• changed

Comments

This sequence shares infinitely many terms with A067742 from which first differs at a(18). It also shares with A067742 the positions of zeros and nonzeros.

Observation: consider the 2-dense sublists of divisors of n. At least for the first 88 terms a(n) coincides with the number of odd terms in the central 2-dense sublist of divisors of n. For more information see A384225 and A280940.

See the "Discussion" text file in the first link for more comments.

Examples

			See the "Discussion" text file in the first link for the examples.
		

Links

• Omar E. Pol, Discussion of the Sequence A384230.
• Omar E. Pol, Illustration of initial terms of A000203 in the pyramid.
• Omar E. Pol, Illustration of initial terms of A001065 in the pyramid.
• Omar E. Pol, Illustration of initial terms of A048050 in the pyramid.
• Omar E. Pol, Illustration of initial terms of A067742 in the pyramid.
• Omar E. Pol, Illustration of initial terms of A224613 (black spiders).
• Omar E. Pol, Illustration of initial terms of A237593 (essentially a template for the Pop-Up pyramid).
• Omar E. Pol, Prism of partitions of 10 and its companion tower (both have the same volume).
• Omar E. Pol, The symmetric representation of sigma(n), n = 1..64 (the top view of the pyramid and of the tower).

Crossrefs

Cf. A001227 (number of subparts), A071561 (positions of zeros), A071562 (positions of nonzeros), A237270 (parts), A237271, A237593, A279387 (subparts), A280940, A384225, A335574, A338488, A377654.

See the "Discussion" text file in the first link for more cross-references.

Formula

a(n) = 0 if and only if A067742(n) = 0.

a(n) >= A067742(n).

(a(n) - A067742(n)) is an even number.

Extensions

Edited by Omar E. Pol, Aug 24 2025