cp's OEIS Frontend

A357636 Numbers k such that the skew-alternating sum of the partition having Heinz number k is 0.

%I A357636 #6 Oct 10 2022 20:47:00
%S A357636 1,4,9,12,16,25,30,36,49,63,64,70,81,90,100,108,121,144,154,165,169,
%T A357636 192,196,210,225,256,273,286,289,300,324,325,360,361,400,441,442,462,
%U A357636 480,484,525,529,550,561,576,588,595,625,646,676,700,729,741,750,784
%N A357636 Numbers k such that the skew-alternating sum of the partition having Heinz number k is 0.
%C A357636 We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ....
%C A357636 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 A357636 The terms together with their prime indices begin:
%e A357636     1: {}
%e A357636     4: {1,1}
%e A357636     9: {2,2}
%e A357636    12: {1,1,2}
%e A357636    16: {1,1,1,1}
%e A357636    25: {3,3}
%e A357636    30: {1,2,3}
%e A357636    36: {1,1,2,2}
%e A357636    49: {4,4}
%e A357636    63: {2,2,4}
%e A357636    64: {1,1,1,1,1,1}
%e A357636    70: {1,3,4}
%e A357636    81: {2,2,2,2}
%e A357636    90: {1,2,2,3}
%e A357636   100: {1,1,3,3}
%e A357636   108: {1,1,2,2,2}
%e A357636   121: {5,5}
%e A357636   144: {1,1,1,1,2,2}
%t A357636 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A357636 skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
%t A357636 Select[Range[1000],skats[Reverse[primeMS[#]]]==0&]
%Y A357636 The version for original alternating sum is A000290.
%Y A357636 The half-alternating form is A000583, non-reverse A357631.
%Y A357636 The version for standard compositions is A357628, non-reverse A357627.
%Y A357636 The non-reverse version is A357632.
%Y A357636 Positions of zeros in A357634, non-reverse A357630.
%Y A357636 These partitions are counted by A357640, half A357639.
%Y A357636 A056239 adds up prime indices, row sums of A112798.
%Y A357636 A316524 gives alternating sum of prime indices, reverse A344616.
%Y A357636 A351005 = alternately equal and unequal partitions, compositions A357643.
%Y A357636 A351006 = alternately unequal and equal partitions, compositions A357644.
%Y A357636 A357641 counts comps w/ half-alt sum 0, even A357642.
%Y A357636 Cf. A003963, A035594, A053251, A055932, A357189, A357485-A357488, A357621-A357626, A357629, A357637, A357638.
%K A357636 nonn
%O A357636 1,2
%A A357636 _Gus Wiseman_, Oct 09 2022