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

1,2

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.

Links

Table of n, a(n) for n=1..50.

Example

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}

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[#]]&]

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.

Sequence in context: A338053 A030661 A245630 * A049728 A038666 A075625

Adjacent sequences: A357845 A357846 A357847 * A357849 A357850 A357851

Keyword

nonn

Author

Gus Wiseman, Oct 16 2022

Status

approved