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

1,8

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.

Links

Table of n, a(n) for n=1..85.

Example

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

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}]

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.

Sequence in context: A256449 A355340 A275327 * A215486 A083721 A378369

Adjacent sequences: A359674 A359675 A359676 * A359678 A359679 A359680

Keyword

nonn

Author

Gus Wiseman, Jan 13 2023

Status

approved