A357633 Half-alternating sum of the partition having Heinz number n.

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

Offset

1,3

Comments

We define the half-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..81.

Example

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

Mathematica

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

Crossrefs

The original alternating sum is A316524, reverse A344616.

The version for standard compositions is A357622, non-reverse A357621.

The skew-alternating form is A357634, non-reverse A357630.

Positions of zeros are A000583, non-reverse A357631.

The reverse version is A357629.

These partitions are counted by A357637, skew A357638.

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, A357623-A357626, A357632, A357636, A357640.

Sequence in context: A071508 A326031 A357629 * A322997 A367315 A085561

Adjacent sequences: A357630 A357631 A357632 * A357634 A357635 A357636

Keyword

nonn

Author

Gus Wiseman, Oct 09 2022

Status

approved