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

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

Author

Gus Wiseman, Oct 01 2022

Keywords

• nonn

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.

Examples

			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}
		

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.

Cf. A051159, A131044, A262046.

Programs

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[#]]]&]