A329321 a(n) is the total number of odd parts in all partitions of n into consecutive parts.

1, 0, 2, 0, 2, 2, 2, 0, 3, 2, 2, 2, 2, 2, 6, 0, 2, 4, 2, 2, 6, 2, 2, 2, 5, 2, 6, 4, 2, 6, 2, 0, 6, 2, 8, 6, 2, 2, 6, 2, 2, 8, 2, 4, 14, 2, 2, 2, 5, 4, 6, 4, 2, 8, 10, 4, 6, 2, 2, 8, 2, 2, 14, 0, 10, 10, 2, 4, 6, 8, 2, 6, 2, 2, 14, 4, 10, 10, 2, 2, 11, 2, 2, 10, 10, 2, 6, 6, 2, 16

Offset

1,3

Comments

a(n) = 0 if and only if n is an even power of 2.

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

a(n) = A204217(n) - A329322(n).

Example

For n = 15 there are four partitions of 15 into consecutive part, they are [15], [8, 7], [6, 5, 4], [5, 4, 3, 2, 1]. In total there are six odd parts, they are [15, 7, 5, 5, 3, 1], so a(15) = 6.

Prog

(PARI) A329321(n) = { my(i=2, t=(n%2)); n--; while(n>0, if(!(n%i), t += (((n/i)%2)+i)\2); n-=i; i++); t }; \\ (After David A. Corneth's program for A204217) - Antti Karttunen, Dec 09 2021

Crossrefs

Cf. A204217 (total number of parts).

Cf. A000079, A001227, A002131, A237048, A245579, A285898, A285914, A286000, A286001, A299765, A328361, A328365, A329322.

Sequence in context: A102395 A127504 A321665 * A334239 A335062 A047917

Adjacent sequences: A329318 A329319 A329320 * A329322 A329323 A329324

Keyword

nonn

Author

Omar E. Pol, Nov 10 2019

Status

approved