A357848 Heinz numbers of integer partitions whose length is twice their alternating sum.

1, 6, 15, 35, 40, 77, 84, 90, 143, 189, 210, 220, 221, 224, 250, 323, 364, 437, 462, 490, 495, 504, 525, 528, 667, 748, 819, 858, 899, 988, 1029, 1040, 1134, 1147, 1155, 1188, 1210, 1320, 1326, 1375, 1400, 1408, 1517, 1564, 1683, 1690, 1763, 1904, 1938, 2021

Offset: 1

Author

Gus Wiseman, Oct 16 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: {}
     6: {1,2}
    15: {2,3}
    35: {3,4}
    40: {1,1,1,3}
    77: {4,5}
    84: {1,1,2,4}
    90: {1,2,2,3}
   143: {5,6}
   189: {2,2,2,4}
   210: {1,2,3,4}
   220: {1,1,3,5}
   221: {6,7}
   224: {1,1,1,1,1,4}
		

Crossrefs

These partitions are counted by A357709.

The version for compositions is counted by A357847.

A000041 counts partitions, strict A000009.

A003963 multiplies prime indices.

A025047 counts alternating compositions.

A056239 adds up prime indices.

A103919 counts partitions by alternating sum, full triangle A344651.

A357136 counts compositions by alternating sum, full triangle A097805.

A357182 counts compositions w/ length = alternating sum, ranked by A357184.

A357189 counts partitions w/ length = alternating sum, ranked by A357486.

Cf. A000720, A001221, A001222, A262977, A301987, A357183, A357485, A357488.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[1000],Length[primeMS[#]]==2sats[primeMS[#]]&]