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A357486
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Heinz numbers of integer partitions with the same length as alternating sum.
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10
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1, 2, 10, 20, 21, 42, 45, 55, 88, 91, 105, 110, 125, 156, 176, 182, 187, 198, 231, 245, 247, 312, 340, 351, 374, 390, 391, 396, 429, 494, 532, 544, 550, 551, 605, 663, 680, 702, 713, 714, 765, 780, 782, 845, 891, 910, 912, 969, 975, 1012, 1064, 1073, 1078
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OFFSET
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1,2
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COMMENTS
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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.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
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LINKS
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EXAMPLE
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The terms together with their prime indices begin:
1: {}
2: {1}
10: {1,3}
20: {1,1,3}
21: {2,4}
42: {1,2,4}
45: {2,2,3}
55: {3,5}
88: {1,1,1,5}
91: {4,6}
105: {2,3,4}
110: {1,3,5}
125: {3,3,3}
156: {1,1,2,6}
176: {1,1,1,1,5}
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]], {i, Length[y]}];
Select[Range[100], PrimeOmega[#]==ats[Reverse[primeMS[#]]]&]
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CROSSREFS
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For product instead of length we have new, counted by A004526.
These partitions are counted by A357189.
A000712 up to 0's counts partitions, sum = twice alt sum, rank A349159.
A001055 counts partitions with product equal to sum, ranked by A301987.
A006330 up to 0's counts partitions, sum = twice rev-alt sum, rank A349160.
A025047 counts alternating compositions.
A357136 counts compositions by alternating sum.
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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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