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A357486 Heinz numbers of integer partitions with the same length as alternating sum.

%I A357486 #6 Oct 02 2022 10:33:39
%S A357486 1,2,10,20,21,42,45,55,88,91,105,110,125,156,176,182,187,198,231,245,
%T A357486 247,312,340,351,374,390,391,396,429,494,532,544,550,551,605,663,680,
%U A357486 702,713,714,765,780,782,845,891,910,912,969,975,1012,1064,1073,1078
%N A357486 Heinz numbers of integer partitions with the same length as alternating sum.
%C A357486 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.
%C A357486 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%e A357486 The terms together with their prime indices begin:
%e A357486      1: {}
%e A357486      2: {1}
%e A357486     10: {1,3}
%e A357486     20: {1,1,3}
%e A357486     21: {2,4}
%e A357486     42: {1,2,4}
%e A357486     45: {2,2,3}
%e A357486     55: {3,5}
%e A357486     88: {1,1,1,5}
%e A357486     91: {4,6}
%e A357486    105: {2,3,4}
%e A357486    110: {1,3,5}
%e A357486    125: {3,3,3}
%e A357486    156: {1,1,2,6}
%e A357486    176: {1,1,1,1,5}
%t A357486 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A357486 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A357486 Select[Range[100],PrimeOmega[#]==ats[Reverse[primeMS[#]]]&]
%Y A357486 For product instead of length we have new, counted by A004526.
%Y A357486 The version for compositions is A357184, counted by A357182.
%Y A357486 For absolute value we have A357486, counted by A357487.
%Y A357486 These partitions are counted by A357189.
%Y A357486 A000041 counts partitions, strict A000009.
%Y A357486 A000712 up to 0's counts partitions, sum = twice alt sum, rank A349159.
%Y A357486 A001055 counts partitions with product equal to sum, ranked by A301987.
%Y A357486 A006330 up to 0's counts partitions, sum = twice rev-alt sum, rank A349160.
%Y A357486 A025047 counts alternating compositions.
%Y A357486 A357136 counts compositions by alternating sum.
%Y A357486 Cf. A051159, A131044, A262046.
%K A357486 nonn
%O A357486 1,2
%A A357486 _Gus Wiseman_, Oct 01 2022