login
Numbers k such that the half-alternating sum of the partition having Heinz number k is 1.
11

%I #6 Oct 29 2022 09:10:03

%S 2,8,24,32,54,128,135,162,375,384,512,648,864,875,1250,1715,1944,2048,

%T 2160,2592,3773,4374,4802,5000,6000,6144,8192,9317,10368,10935,13122,

%U 13824,14000,15000,17303,19208,20000,24167,27440,29282,30375,31104,32768,33750

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

%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 terms together with their prime indices begin:

%e 2: {1}

%e 8: {1,1,1}

%e 24: {1,1,1,2}

%e 32: {1,1,1,1,1}

%e 54: {1,2,2,2}

%e 128: {1,1,1,1,1,1,1}

%e 135: {2,2,2,3}

%e 162: {1,2,2,2,2}

%e 375: {2,3,3,3}

%e 384: {1,1,1,1,1,1,1,2}

%e 512: {1,1,1,1,1,1,1,1,1}

%e 648: {1,1,1,2,2,2,2}

%e 864: {1,1,1,1,1,2,2,2}

%e 875: {3,3,3,4}

%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 Select[Range[1000],halfats[Reverse[primeMS[#]]]==1&]

%Y The version for k = 0 is A000583, standard compositions A357625-A357626.

%Y The version for original alternating sum is A345958.

%Y Positions of ones in A357633, non-reverse A357629.

%Y The skew version for k = 0 is A357636, non-reverse A357632.

%Y These partitions are counted by A035444, skew A035544.

%Y The non-reverse version is A357851, k = 0 version A357631.

%Y A056239 adds up prime indices, row sums of A112798.

%Y A316524 gives alternating sum of prime indices, reverse A344616.

%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, even-length A357642.

%Y Cf. A000290, A003963, A053251, A055932, A357621-A357624, A357630, A357634, A357637, A357639, A357640.

%K nonn

%O 1,1

%A _Gus Wiseman_, Oct 28 2022