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 A357636 Numbers k such that the skew-alternating sum of the partition having Heinz number k is 0. 17
 1, 4, 9, 12, 16, 25, 30, 36, 49, 63, 64, 70, 81, 90, 100, 108, 121, 144, 154, 165, 169, 192, 196, 210, 225, 256, 273, 286, 289, 300, 324, 325, 360, 361, 400, 441, 442, 462, 480, 484, 525, 529, 550, 561, 576, 588, 595, 625, 646, 676, 700, 729, 741, 750, 784 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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 + .... 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. LINKS Table of n, a(n) for n=1..55. EXAMPLE The terms together with their prime indices begin: 1: {} 4: {1,1} 9: {2,2} 12: {1,1,2} 16: {1,1,1,1} 25: {3,3} 30: {1,2,3} 36: {1,1,2,2} 49: {4,4} 63: {2,2,4} 64: {1,1,1,1,1,1} 70: {1,3,4} 81: {2,2,2,2} 90: {1,2,2,3} 100: {1,1,3,3} 108: {1,1,2,2,2} 121: {5,5} 144: {1,1,1,1,2,2} MATHEMATICA primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]; skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]), {i, Length[f]}]; Select[Range[1000], skats[Reverse[primeMS[#]]]==0&] CROSSREFS The version for original alternating sum is A000290. The half-alternating form is A000583, non-reverse A357631. The version for standard compositions is A357628, non-reverse A357627. The non-reverse version is A357632. Positions of zeros in A357634, non-reverse A357630. These partitions are counted by A357640, half A357639. A056239 adds up prime indices, row sums of A112798. A316524 gives alternating sum of prime indices, reverse A344616. A351005 = alternately equal and unequal partitions, compositions A357643. A351006 = alternately unequal and equal partitions, compositions A357644. A357641 counts comps w/ half-alt sum 0, even A357642. Cf. A003963, A035594, A053251, A055932, A357189, A357485-A357488, A357621-A357626, A357629, A357637, A357638. Sequence in context: A320924 A357976 A330879 * A363261 A360953 A348272 Adjacent sequences: A357633 A357634 A357635 * A357637 A357638 A357639 KEYWORD nonn AUTHOR Gus Wiseman, Oct 09 2022 STATUS approved

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Last modified August 8 12:42 EDT 2024. Contains 375021 sequences. (Running on oeis4.)