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A357635 Numbers k such that the half-alternating sum of the partition having Heinz number k is 1.

%I A357635 #6 Oct 29 2022 09:10:03
%S A357635 2,8,24,32,54,128,135,162,375,384,512,648,864,875,1250,1715,1944,2048,
%T A357635 2160,2592,3773,4374,4802,5000,6000,6144,8192,9317,10368,10935,13122,
%U A357635 13824,14000,15000,17303,19208,20000,24167,27440,29282,30375,31104,32768,33750
%N A357635 Numbers k such that the half-alternating sum of the partition having Heinz number k is 1.
%C A357635 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 A357635 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 A357635 The terms together with their prime indices begin:
%e A357635     2: {1}
%e A357635     8: {1,1,1}
%e A357635    24: {1,1,1,2}
%e A357635    32: {1,1,1,1,1}
%e A357635    54: {1,2,2,2}
%e A357635   128: {1,1,1,1,1,1,1}
%e A357635   135: {2,2,2,3}
%e A357635   162: {1,2,2,2,2}
%e A357635   375: {2,3,3,3}
%e A357635   384: {1,1,1,1,1,1,1,2}
%e A357635   512: {1,1,1,1,1,1,1,1,1}
%e A357635   648: {1,1,1,2,2,2,2}
%e A357635   864: {1,1,1,1,1,2,2,2}
%e A357635   875: {3,3,3,4}
%t A357635 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A357635 halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
%t A357635 Select[Range[1000],halfats[Reverse[primeMS[#]]]==1&]
%Y A357635 The version for k = 0 is A000583, standard compositions A357625-A357626.
%Y A357635 The version for original alternating sum is A345958.
%Y A357635 Positions of ones in A357633, non-reverse A357629.
%Y A357635 The skew version for k = 0 is A357636, non-reverse A357632.
%Y A357635 These partitions are counted by A035444, skew A035544.
%Y A357635 The non-reverse version is A357851, k = 0 version A357631.
%Y A357635 A056239 adds up prime indices, row sums of A112798.
%Y A357635 A316524 gives alternating sum of prime indices, reverse A344616.
%Y A357635 A351005 = alternately equal and unequal partitions, compositions A357643.
%Y A357635 A351006 = alternately unequal and equal partitions, compositions A357644.
%Y A357635 A357641 counts comps w/ half-alt sum 0, even-length A357642.
%Y A357635 Cf. A000290, A003963, A053251, A055932, A357621-A357624, A357630, A357634, A357637, A357639, A357640.
%K A357635 nonn
%O A357635 1,1
%A A357635 _Gus Wiseman_, Oct 28 2022