A359677 - OEIS

%I #5 Jan 15 2023 09:51:09

%S 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 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 3,2,5,0,11,0,1,7,3,4,4,0,10,12,1,0,7,3

%N Zero-based weighted sum of the reversed (weakly decreasing) 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 The zero-based weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} (i-1)*y_i.

%e The reversed prime indices of 12 are (2,1,1), so a(12) = 0*2 + 1*1 + 2*1 = 3.

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t wts[y_]:=Sum[(i-1)*y[[i]],{i,Length[y]}];

%t Table[wts[Reverse[primeMS[n]]],{n,100}]

%Y Positions of 0's are A008578.

%Y Positions of 1's are A100484.

%Y The version for standard compositions is A231204, reverse of A124757.

%Y The one-based version is A318283, unreversed A304818.

%Y The one-based version for standard compositions is A359042, rev of A029931.

%Y This is the reverse version of A359674.

%Y First position of n is A359679(n), reverse of A359675.

%Y Positions of first appearances are A359680, reverse of A359676.

%Y A053632 counts compositions by weighted sum.

%Y A112798 lists prime indices, length A001222, sum A056239.

%Y A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.

%Y Cf. A001248, A055932, A243055, A359043, A358194, A359678, A359681, A359683, A359754, A359755.

%K nonn

%O 1,8

%A _Gus Wiseman_, Jan 13 2023