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%I #8 Oct 24 2023 18:46:45
%S 0,-1,2,-2,-3,1,4,-3,4,-4,-5,0,6,3,-1,-4,-7,3,8,-5,6,-6,-9,-1,-6,5,6,
%T 2,10,-2,-11,-5,-3,-8,1,2,12,7,8,-6,-13,5,14,-7,1,-10,-15,-2,8,-7,-5,
%U 4,16,5,-8,1,10,9,-17,-3,18,-12,8,-6,3,-4,-19,-9,-7,0
%N Self-signed alternating sum of the prime indices of n.
%C We define the self-signed alternating sum of a multiset y to be Sum_{k in y} k*(-1)^k.
%C 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.
%F a(n) = Sum_{k in A112798(n)} k*(-1)^k.
%F a(n) = A366531(n) - A366528(n).
%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t asum[y_]:=Sum[x*(-1)^x,{x,y}];
%t Table[asum[prix[n]],{n,100}]
%Y With summands of 2^(n-1) we get A048675.
%Y With summands of (-1)^k we get A195017.
%Y The version for alternating prime indices is A346697 - A346698 = A316524.
%Y Positions of zeros are A366748, counted by A239261.
%Y A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
%Y A300061 lists numbers with even sum of prime indices, odd A300063.
%Y A366528 adds up odd prime indices, counted by A113685.
%Y A366531 adds up even prime indices, counted by A113686.
%Y Cf. A000720, A019507, A045931, A055396, A061395, A325698, A325700, A366533.
%K sign
%O 1,3
%A _Gus Wiseman_, Oct 23 2023
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