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A357486
Heinz numbers of integer partitions with the same length as alternating sum.
10
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
OFFSET
1,2
COMMENTS
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.
EXAMPLE
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}
MATHEMATICA
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[#]]]&]
CROSSREFS
For product instead of length we have new, counted by A004526.
The version for compositions is A357184, counted by A357182.
For absolute value we have A357486, counted by A357487.
These partitions are counted by A357189.
A000041 counts partitions, strict A000009.
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.
Sequence in context: A346810 A347024 A009342 * A350508 A306105 A038103
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 01 2022
STATUS
approved