

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
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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)^(i1) y_i.


LINKS



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)^(i1)*y[[i]], {i, Length[y]}];
Select[Range[100], PrimeOmega[#]==ats[Reverse[primeMS[#]]]&]


CROSSREFS

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 revalt sum, rank A349160.
A025047 counts alternating compositions.
A357136 counts compositions by alternating sum.


KEYWORD

nonn


AUTHOR



STATUS

approved



