A329321 - OEIS

%I #24 Dec 10 2021 11:37:47

%S 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,

%T 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,

%U 2,4,6,8,2,6,2,2,14,4,10,10,2,2,11,2,2,10,10,2,6,6,2,16

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

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

%H Antti Karttunen, <a href="/A329321/b329321.txt">Table of n, a(n) for n = 1..20000</a>

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

%e 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.

%o (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

%Y Cf. A204217 (total number of parts).

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

%K nonn

%O 1,3

%A _Omar E. Pol_, Nov 10 2019