%I #6 Oct 02 2022 10:33:45
%S 1,2,20,42,45,105,110,125,176,182,231,245,312,374,396,429,494,605,663,
%T 680,702,780,782,845,891,969,1064,1088,1100,1102,1311,1426,1428,1445,
%U 1530,1755,1805,1820,1824,1950,2001,2024,2146,2156,2394,2448,2475,2508,2542
%N Heinz numbers of integer partitions with the same length as reverse-alternating sum.
%C 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 The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^i y_i.
%e The terms together with their prime indices begin:
%e 1: {}
%e 2: {1}
%e 20: {1,1,3}
%e 42: {1,2,4}
%e 45: {2,2,3}
%e 105: {2,3,4}
%e 110: {1,3,5}
%e 125: {3,3,3}
%e 176: {1,1,1,1,5}
%e 182: {1,4,6}
%e 231: {2,4,5}
%e 245: {3,4,4}
%e 312: {1,1,1,2,6}
%e 374: {1,5,7}
%e 396: {1,1,2,2,5}
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t Select[Range[100],PrimeOmega[#]==ats[primeMS[#]]&]
%Y The version for compositions is A357184, counted by A357182.
%Y These partitions are counted by A357189.
%Y For absolute value we have A357486, counted by A357487.
%Y A000041 counts partitions, strict A000009.
%Y A000712 up to 0's counts partitions w sum = twice alt sum, ranked A349159.
%Y A001055 counts partitions with product equal to sum, ranked by A301987.
%Y A006330 up to 0's counts partitions w sum = twice rev-alt sum, rank A349160.
%Y Cf. A004526, A025047, A051159, A131044, A262046, A357136.
%K nonn
%O 1,2
%A _Gus Wiseman_, Oct 01 2022