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%I #14 Aug 16 2023 10:59:06
%S 0,1,2,-1,3,-3,4,2,-2,-5,5,5,6,-7,-4,-2,7,3,8,8,-6,-9,9,-6,-3,-11,4,
%T 11,10,6,11,3,-8,-13,-5,-3,12,-15,-10,-10,13,9,14,14,7,-17,15,8,-4,4,
%U -12,17,16,-5,-7,-14,-14,-19,17,-7,18,-21,10,-3,-9,12,19,20
%N Weighted alternating sum of the multiset of prime indices of n.
%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.
%C We define the weighted alternating sum of a sequence (y_1,...,y_k) to be Sum_{i=1..k} (-1)^(i-1) i * y_i.
%e The prime indices of 300 are {1,1,2,3,3}, with weighted alternating sum 1*1 - 2*1 + 3*2 - 4*3 + 5*3 = 8, so a(300) = 8.
%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t altwtsum[y_]:=Sum[(-1)^(k-1)*k*y[[k]],{k,1,Length[y]}];
%t Table[altwtsum[prix[n]],{n,100}]
%Y The non-alternating version is A304818, reverse A318283.
%Y The unweighted version is A316524, reverse A344616.
%Y The reverse version is A363620.
%Y The triangle for this rank statistic is A363622, reverse A363623.
%Y For partitions instead of multisets we have A363624, reverse A363625.
%Y A055396 gives minimum prime index, maximum A061395.
%Y A112798 lists prime indices, length A001222, sum A056239.
%Y A264034 counts partitions by weighted sum, reverse A358194.
%Y A320387 counts multisets by weighted sum, zero-based A359678.
%Y A359677 gives zero-based weighted sum of prime indices, reverse A359674.
%Y Cf. A000009, A000720, A001221, A046660, A053632, A106529, A124010, A181819, A261079, A363532, A363621, A363626.
%K sign
%O 1,3
%A _Gus Wiseman_, Jun 12 2023
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