A366533 Sum of even prime indices of n divided by 2.

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

Offset: 1

Author

Gus Wiseman, Oct 23 2023

Keywords

• nonn

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The prime indices of 198 are {1,2,2,5}, so a(198) = (2+2)/2 = 2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Zeros are A066208, counted by A000009.

The triangle for this statistic (without zeros) is A174713.

The un-halved odd version is A366528.

The un-halved version is A366531.

A066207 lists numbers with all even prime indices, counted by A035363.

A112798 lists prime indices, reverse A296150, length A001222, sum A056239.

A113685 counts partitions by sum of odd parts, even version A113686.

A239261 counts partitions with (sum of odd parts) = (sum of even parts).

A257991 counts odd prime indices, even A257992.

A346697 adds up odd-indexed prime indices, even-indexed A346698.

A365067 counts partitions by sum of odd parts (without zeros).

A366322 lists numbers with not all prime indices even, counted by A086543.

Cf. A000720, A055396, A055922, A061395, A162641, A171966, A258117, A325698, A325700, A352140, A352141.

Programs

Maple

f:= proc(n) local F,t;
  F:= map(t -> [numtheory:-Pi(t[1]),t[2]], ifactors(n)[2]);
  add(`if`(t[1]::even, t[1]*t[2]/2, 0), t=F)
end proc:
map(f, [$1..100]); # Robert Israel, Nov 22 2023

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[Select[prix[n],EvenQ]]/2,{n,100}]

Formula

a(n) = A366531(n)/2.