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).

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

Offset: 1

Author

Antti Karttunen, Jun 05 2014

Keywords

• nonn

Comments

Each n occurs A000041(n) times in total.

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

Links

• Antti Karttunen, Table of n, a(n) for n = 1..8192

Crossrefs

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

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

Sums of even-indexed rows of A358172.

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

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

Programs

Mathematica

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 *)

Formula

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

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

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

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

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

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

a(n) = A061395(n) * A001222(n) + A061395(n) - A056239(n) + A001222(n) - 1. - Gus Wiseman, Jan 09 2023

a(n) = A326844(2n) + A001222(n). - Gus Wiseman, Jan 09 2023