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A243503 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).

%I A243503 #23 Jan 11 2023 06:40:33
%S A243503 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 A243503 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 A243503 20,17,11,18,22,10,6,10,14,19,21,17,10,20,9,21,24,6,24
%N A243503 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 A243503 Each n occurs A000041(n) times in total.
%C A243503 Where are the first and the last occurrence of each n located?
%H A243503 Antti Karttunen, <a href="/A243503/b243503.txt">Table of n, a(n) for n = 1..8192</a>
%F A243503 a(n) = Sum_{i=A203623(n-1)+2..A203623(n)+1} A241918(i).
%F A243503 a(n) = A056239(A241909(n)).
%F A243503 a(n) = A227183(A075158(n-1)).
%F A243503 a(A000040(n)) = a(A000079(n)) = n for all n >= 1.
%F A243503 a(A122111(n)) = a(n) for all n.
%F A243503 a(A243051(n)) = a(n) for all n, and likewise for A243052, A243053 and other rows of A243060.
%F A243503 a(n) = A061395(n) * A001222(n) + A061395(n) - A056239(n) + A001222(n) - 1. - _Gus Wiseman_, Jan 09 2023
%F A243503 a(n) = A326844(2n) + A001222(n). - _Gus Wiseman_, Jan 09 2023
%t A243503 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 A243503 Cf. A243504 (the products of parts), A241918, A000041, A227183, A075158, A056239, A241909.
%Y A243503 Sum of prime indices of A241916, the even bisection of A358195.
%Y A243503 Sums of even-indexed rows of A358172.
%Y A243503 A112798 lists prime indices, length A001222, sum A056239, max A061395.
%Y A243503 Cf. A005940, A019565, A246277, A326844, A356958, A359362.
%K A243503 nonn
%O A243503 1,3
%A A243503 _Antti Karttunen_, Jun 05 2014