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

Author

Gus Wiseman, Oct 09 2022

Keywords

• nonn

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.

Examples

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

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.

Programs

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}]