A357634 Skew-alternating sum of the partition having Heinz number n.

0, 1, 2, 0, 3, 1, 4, -1, 0, 2, 5, 0, 6, 3, 1, 0, 7, -1, 8, 1, 2, 4, 9, 1, 0, 5, -2, 2, 10, 0, 11, 1, 3, 6, 1, 0, 12, 7, 4, 2, 13, 1, 14, 3, -1, 8, 15, 2, 0, -1, 5, 4, 16, -1, 2, 3, 6, 9, 17, 1, 18, 10, 0, 0, 3, 2, 19, 5, 7, 0, 20, 1, 21, 11, -2, 6, 1, 3, 22, 3

Offset

1,3

Comments

We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ....

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.

Links

Table of n, a(n) for n=1..80.

Example

The partition with Heinz number 525 is (4,3,3,2) so a(525) = 4 - 3 - 3 + 2 = 0.

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]), {i, Length[f]}];
Table[skats[Reverse[primeMS[n]]], {n, 30}]

Crossrefs

The original alternating sum is A316524, reverse A344616.

The non-reverse version is A357630.

The half-alternating form is A357633, non-reverse A357629.

Positions of zeros are A357636, non-reverse A357632.

The version for standard compositions is A357624, non-reverse A357623.

These partitions are counted by A357638, half A357637.

A056239 adds up prime indices, row sums of A112798.

A351005 = alternately equal and unequal partitions, compositions A357643.

A351006 = alternately unequal and equal partitions, compositions A357644.

A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.

Cf. A003963, A053251, A055932, A357189, A357485-A357488, A357621, A357622, A357625, A357626, A357635, A357640.

Sequence in context: A328967 A358171 A277707 * A344616 A316524 A357630

Adjacent sequences: A357631 A357632 A357633 * A357635 A357636 A357637

Keyword

sign

Author

Gus Wiseman, Oct 09 2022

Status

approved