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A357636
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Numbers k such that the skew-alternating sum of the partition having Heinz number k is 0.
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17
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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}
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MATHEMATICA
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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&]
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CROSSREFS
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The version for original alternating sum is A000290.
The version for standard compositions is A357628, non-reverse A357627.
The non-reverse version is A357632.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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