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A357635
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Numbers k such that the half-alternating sum of the partition having Heinz number k is 1.
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11
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2, 8, 24, 32, 54, 128, 135, 162, 375, 384, 512, 648, 864, 875, 1250, 1715, 1944, 2048, 2160, 2592, 3773, 4374, 4802, 5000, 6000, 6144, 8192, 9317, 10368, 10935, 13122, 13824, 14000, 15000, 17303, 19208, 20000, 24167, 27440, 29282, 30375, 31104, 32768, 33750
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OFFSET
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1,1
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COMMENTS
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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 - ...
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:
2: {1}
8: {1,1,1}
24: {1,1,1,2}
32: {1,1,1,1,1}
54: {1,2,2,2}
128: {1,1,1,1,1,1,1}
135: {2,2,2,3}
162: {1,2,2,2,2}
375: {2,3,3,3}
384: {1,1,1,1,1,1,1,2}
512: {1,1,1,1,1,1,1,1,1}
648: {1,1,1,2,2,2,2}
864: {1,1,1,1,1,2,2,2}
875: {3,3,3,4}
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]), {i, Length[f]}];
Select[Range[1000], halfats[Reverse[primeMS[#]]]==1&]
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CROSSREFS
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The version for original alternating sum is A345958.
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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