login
Heinz numbers of integer partitions with the same length as reverse-alternating sum.
13

%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