cp's OEIS Frontend

A363620 Reverse-weighted alternating sum of the multiset of prime indices of n.

%I A363620 #13 Aug 16 2023 10:59:23
%S A363620 0,1,2,1,3,0,4,2,2,-1,5,3,6,-2,1,2,7,1,8,4,0,-3,9,1,3,-4,4,5,10,2,11,
%T A363620 3,-1,-5,2,3,12,-6,-2,0,13,3,14,6,5,-7,15,4,4,0,-3,7,16,0,1,-1,-4,-8,
%U A363620 17,2,18,-9,6,3,0,4,19,8,-5,1,20,2,21,-10,3,9,3
%N A363620 Reverse-weighted alternating sum of the multiset of prime indices of n.
%C A363620 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 A363620 We define the reverse-weighted alternating sum of a sequence (y_1,...,y_k) to be Sum_{i=1..k} (-1)^(k-i) i * y_{k-i+1}.
%e A363620 The prime indices of 300 are {1,1,2,3,3}, with reverse-weighted alternating sum 1*3 - 2*3 + 3*2 - 4*1 + 5*1 = 4, so a(300) = 4.
%t A363620 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A363620 revaltwtsum[y_]:=Sum[(-1)^(Length[y]-k)*k*y[[-k]],{k,1,Length[y]}];
%t A363620 Table[revaltwtsum[prix[n]],{n,100}]
%Y A363620 The reverse non-alternating version is A304818, row-sums of A359361.
%Y A363620 The non-alternating version is A318283, row-sums of A358136.
%Y A363620 The unweighted version is A344616, reverse A316524.
%Y A363620 The reverse version is A363619.
%Y A363620 Positions of zeros are A363621.
%Y A363620 The triangle for this rank statistic is A363623, reverse A363622.
%Y A363620 For partitions instead of multisets we have A363625, reverse A363624.
%Y A363620 A055396 gives minimum prime index, maximum A061395.
%Y A363620 A112798 lists prime indices, length A001222, sum A056239.
%Y A363620 A264034 counts partitions by weighted sum, reverse A358194.
%Y A363620 A320387 counts multisets by weighted sum, zero-based A359678.
%Y A363620 Cf. A001221, A046660, A053632, A106529, A124010, A222855, A261079, A358137, A359674, A359755, A363531, A363532.
%K A363620 sign
%O A363620 1,3
%A A363620 _Gus Wiseman_, Jun 13 2023