A363619 Weighted alternating sum of the multiset of prime indices of n.

0, 1, 2, -1, 3, -3, 4, 2, -2, -5, 5, 5, 6, -7, -4, -2, 7, 3, 8, 8, -6, -9, 9, -6, -3, -11, 4, 11, 10, 6, 11, 3, -8, -13, -5, -3, 12, -15, -10, -10, 13, 9, 14, 14, 7, -17, 15, 8, -4, 4, -12, 17, 16, -5, -7, -14, -14, -19, 17, -7, 18, -21, 10, -3, -9, 12, 19, 20

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 weighted alternating sum of a sequence (y_1,...,y_k) to be Sum_{i=1..k} (-1)^(i-1) i * y_i.

Links

Table of n, a(n) for n=1..68.

Example

The prime indices of 300 are {1,1,2,3,3}, with weighted alternating sum 1*1 - 2*1 + 3*2 - 4*3 + 5*3 = 8, so a(300) = 8.

Mathematica

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

Crossrefs

The non-alternating version is A304818, reverse A318283.

The unweighted version is A316524, reverse A344616.

The reverse version is A363620.

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

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

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.

A359677 gives zero-based weighted sum of prime indices, reverse A359674.

Cf. A000009, A000720, A001221, A046660, A053632, A106529, A124010, A181819, A261079, A363532, A363621, A363626.

Sequence in context: A308057 A097686 A082666 * A363625 A057938 A375729

Adjacent sequences: A363616 A363617 A363618 * A363620 A363621 A363622

Keyword

sign

Author

Gus Wiseman, Jun 12 2023

Status

approved