%I #23 Jan 11 2023 06:40:33

%S 0,1,2,2,3,4,4,3,3,6,5,6,6,8,5,4,7,5,8,9,7,10,9,8,4,12,4,12,10,8,11,5,

%T 9,14,6,7,12,16,11,12,13,11,14,15,7,18,15,10,5,7,13,18,16,6,8,16,15,

%U 20,17,11,18,22,10,6,10,14,19,21,17,10,20,9,21,24,6,24

%N Sums of parts of partitions (i.e., their sizes) as ordered in the table A241918: a(n) = Sum_{i=A203623(n-1)+2..A203623(n)+1} A241918(i).

%C Each n occurs A000041(n) times in total.

%C Where are the first and the last occurrence of each n located?

%H Antti Karttunen, <a href="/A243503/b243503.txt">Table of n, a(n) for n = 1..8192</a>

%F a(n) = Sum_{i=A203623(n-1)+2..A203623(n)+1} A241918(i).

%F a(n) = A056239(A241909(n)).

%F a(n) = A227183(A075158(n-1)).

%F a(A000040(n)) = a(A000079(n)) = n for all n >= 1.

%F a(A122111(n)) = a(n) for all n.

%F a(A243051(n)) = a(n) for all n, and likewise for A243052, A243053 and other rows of A243060.

%F a(n) = A061395(n) * A001222(n) + A061395(n) - A056239(n) + A001222(n) - 1. - _Gus Wiseman_, Jan 09 2023

%F a(n) = A326844(2n) + A001222(n). - _Gus Wiseman_, Jan 09 2023

%t Table[If[n==1,0,With[{y=Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]},Last[y]*Length[y]+Last[y]-Total[y]+Length[y]-1]],{n,100}] (* _Gus Wiseman_, Jan 09 2023 *)

%Y Cf. A243504 (the products of parts), A241918, A000041, A227183, A075158, A056239, A241909.

%Y Sum of prime indices of A241916, the even bisection of A358195.

%Y Sums of even-indexed rows of A358172.

%Y A112798 lists prime indices, length A001222, sum A056239, max A061395.

%Y Cf. A005940, A019565, A246277, A326844, A356958, A359362.

%K nonn

%O 1,3

%A _Antti Karttunen_, Jun 05 2014