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

%I A366533 #11 Nov 22 2023 12:09:26
%S A366533 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,
%T A366533 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,
%U A366533 1,2,10,2,0,6,1,4,2,4,11,0,4,0,0,3,0,7
%N A366533 Sum of even prime indices of n divided by 2.
%C A366533 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.
%H A366533 Robert Israel, <a href="/A366533/b366533.txt">Table of n, a(n) for n = 1..10000</a>
%F A366533 a(n) = A366531(n)/2.
%e A366533 The prime indices of 198 are {1,2,2,5}, so a(198) = (2+2)/2 = 2.
%p A366533 f:= proc(n) local F,t;
%p A366533   F:= map(t -> [numtheory:-Pi(t[1]),t[2]], ifactors(n)[2]);
%p A366533   add(`if`(t[1]::even, t[1]*t[2]/2, 0), t=F)
%p A366533 end proc:
%p A366533 map(f, [$1..100]); # _Robert Israel_, Nov 22 2023
%t A366533 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A366533 Table[Total[Select[prix[n],EvenQ]]/2,{n,100}]
%Y A366533 Zeros are A066208, counted by A000009.
%Y A366533 The triangle for this statistic (without zeros) is A174713.
%Y A366533 The un-halved odd version is A366528.
%Y A366533 The un-halved version is A366531.
%Y A366533 A066207 lists numbers with all even prime indices, counted by A035363.
%Y A366533 A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
%Y A366533 A113685 counts partitions by sum of odd parts, even version A113686.
%Y A366533 A239261 counts partitions with (sum of odd parts) = (sum of even parts).
%Y A366533 A257991 counts odd prime indices, even A257992.
%Y A366533 A346697 adds up odd-indexed prime indices, even-indexed A346698.
%Y A366533 A365067 counts partitions by sum of odd parts (without zeros).
%Y A366533 A366322 lists numbers with not all prime indices even, counted by A086543.
%Y A366533 Cf. A000720, A055396, A055922, A061395, A162641, A171966, A258117, A325698, A325700, A352140, A352141.
%K A366533 nonn
%O A366533 1,7
%A A366533 _Gus Wiseman_, Oct 23 2023