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

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

Offset: 1

Author

Gus Wiseman, Jan 13 2023

Keywords

• nonn

Comments

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.

The zero-based weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} (i-1)*y_i.

Examples

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

Crossrefs

Positions of 0's are A008578.

Positions of 1's are A100484.

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

The one-based version is A318283, unreversed A304818.

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

This is the reverse version of A359674.

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

Positions of first appearances are A359680, reverse of A359676.

A053632 counts compositions by weighted sum.

A112798 lists prime indices, length A001222, sum A056239.

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

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

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
wts[y_]:=Sum[(i-1)*y[[i]],{i,Length[y]}];
Table[wts[Reverse[primeMS[n]]],{n,100}]