cp's OEIS Frontend

A359677 Zero-based weighted sum of the reversed (weakly decreasing) prime indices of n.

%I A359677 #5 Jan 15 2023 09:51:09
%S A359677 0,0,0,1,0,1,0,3,2,1,0,3,0,1,2,6,0,4,0,3,2,1,0,6,3,1,6,3,0,4,0,10,2,1,
%T A359677 3,7,0,1,2,6,0,4,0,3,6,1,0,10,4,5,2,3,0,9,3,6,2,1,0,7,0,1,6,15,3,4,0,
%U A359677 3,2,5,0,11,0,1,7,3,4,4,0,10,12,1,0,7,3
%N A359677 Zero-based weighted sum of the reversed (weakly decreasing) prime indices of n.
%C A359677 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 A359677 The zero-based weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} (i-1)*y_i.
%e A359677 The reversed prime indices of 12 are (2,1,1), so a(12) = 0*2 + 1*1 + 2*1 = 3.
%t A359677 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A359677 wts[y_]:=Sum[(i-1)*y[[i]],{i,Length[y]}];
%t A359677 Table[wts[Reverse[primeMS[n]]],{n,100}]
%Y A359677 Positions of 0's are A008578.
%Y A359677 Positions of 1's are A100484.
%Y A359677 The version for standard compositions is A231204, reverse of A124757.
%Y A359677 The one-based version is A318283, unreversed A304818.
%Y A359677 The one-based version for standard compositions is A359042, rev of A029931.
%Y A359677 This is the reverse version of A359674.
%Y A359677 First position of n is A359679(n), reverse of A359675.
%Y A359677 Positions of first appearances are A359680, reverse of A359676.
%Y A359677 A053632 counts compositions by weighted sum.
%Y A359677 A112798 lists prime indices, length A001222, sum A056239.
%Y A359677 A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.
%Y A359677 Cf. A001248, A055932, A243055, A359043, A358194, A359678, A359681, A359683, A359754, A359755.
%K A359677 nonn
%O A359677 1,8
%A A359677 _Gus Wiseman_, Jan 13 2023