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%I #5 Oct 09 2022 20:25:11
%S 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,
%T 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,
%U 4,2,9,6,19,7,11,6,20,3,21,13,4,8,9,7,22,3,0
%N Half-alternating sum of the partition having Heinz number n.
%C 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 - ...
%C 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.
%e The partition with Heinz number 525 is (4,3,3,2) so a(525) = 4 + 3 - 3 - 2 = 2.
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
%t Table[halfats[Reverse[primeMS[n]]],{n,30}]
%Y The original alternating sum is A316524, reverse A344616.
%Y The version for standard compositions is A357622, non-reverse A357621.
%Y The skew-alternating form is A357634, non-reverse A357630.
%Y Positions of zeros are A000583, non-reverse A357631.
%Y The reverse version is A357629.
%Y These partitions are counted by A357637, skew A357638.
%Y A056239 adds up prime indices, row sums of A112798.
%Y A351005 = alternately equal and unequal partitions, compositions A357643.
%Y A351006 = alternately unequal and equal partitions, compositions A357644.
%Y A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.
%Y Cf. A003963, A053251, A055932, A357189, A357485-A357488, A357623-A357626, A357632, A357636, A357640.
%K nonn
%O 1,3
%A _Gus Wiseman_, Oct 09 2022
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