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

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, 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, 17, 2, 18, -9, 6, 3, 0, 4, 19, 8, -5, 1, 20, 2, 21, -10, 3, 9, 3

Offset

1,3

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.

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}.

Links

Table of n, a(n) for n=1..77.

Example

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.

Mathematica

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

Crossrefs

The reverse non-alternating version is A304818, row-sums of A359361.

The non-alternating version is A318283, row-sums of A358136.

The unweighted version is A344616, reverse A316524.

The reverse version is A363619.

Positions of zeros are A363621.

The triangle for this rank statistic is A363623, reverse A363622.

For partitions instead of multisets we have A363625, reverse A363624.

A055396 gives minimum prime index, maximum A061395.

A112798 lists prime indices, length A001222, sum A056239.

A264034 counts partitions by weighted sum, reverse A358194.

A320387 counts multisets by weighted sum, zero-based A359678.

Cf. A001221, A046660, A053632, A106529, A124010, A222855, A261079, A358137, A359674, A359755, A363531, A363532.

Sequence in context: A334594 A359336 A062778 * A363624 A108202 A025480

Adjacent sequences: A363617 A363618 A363619 * A363621 A363622 A363623

Keyword

sign

Author

Gus Wiseman, Jun 13 2023

Status

approved