A363625 Reverse-weighted alternating sum of the integer partition with Heinz number 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, 16, 7, 20, 4, 21, 23, 5, 23

Offset

1,3

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

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..76.

Example

The partition with Heinz number 600 is (3,3,2,1,1,1), so a(600) = -1*1 + 2*1 - 3*1 + 4*2 - 5*3 + 6*3 = 9.

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[Reverse[prix[n]]], {n, 100}]

Crossrefs

The non-alternating version is A304818, reverse A318283.

The unweighted version is A316524, reverse A344616.

For multisets instead of partitions we have A363620.

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

The reverse version is A363624, for multisets A363619.

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, reverse A007294.

Cf. A046660, A053632, A106529, A124010, A261079, A359361, A359755, A363532, A363621, A363626.

Sequence in context: A097686 A082666 A363619 * A057938 A144623 A340284

Adjacent sequences: A363622 A363623 A363624 * A363626 A363627 A363628

Keyword

nonn

Author

Gus Wiseman, Jun 15 2023

Status

approved