A357485 - OEIS

A357485

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

1, 2, 20, 42, 45, 105, 110, 125, 176, 182, 231, 245, 312, 374, 396, 429, 494, 605, 663, 680, 702, 780, 782, 845, 891, 969, 1064, 1088, 1100, 1102, 1311, 1426, 1428, 1445, 1530, 1755, 1805, 1820, 1824, 1950, 2001, 2024, 2146, 2156, 2394, 2448, 2475, 2508, 2542

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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 reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^i y_i.

LINKS

Table of n, a(n) for n=1..49.

EXAMPLE

The terms together with their prime indices begin:

1: {}

2: {1}

20: {1,1,3}

42: {1,2,4}

45: {2,2,3}

105: {2,3,4}

110: {1,3,5}

125: {3,3,3}

176: {1,1,1,1,5}

182: {1,4,6}

231: {2,4,5}

245: {3,4,4}

312: {1,1,1,2,6}

374: {1,5,7}

396: {1,1,2,2,5}

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[primeMS[#]]&]

CROSSREFS

The version for compositions is A357184, counted by A357182.

These partitions are counted by A357189.

For absolute value we have A357486, counted by A357487.

A000041 counts partitions, strict A000009.

A000712 up to 0's counts partitions w sum = twice alt sum, ranked A349159.

A001055 counts partitions with product equal to sum, ranked by A301987.

A006330 up to 0's counts partitions w sum = twice rev-alt sum, rank A349160.

Cf. A004526, A025047, A051159, A131044, A262046, A357136.

Sequence in context: A293356 A269669 A293930 * A194999 A280436 A266842

Adjacent sequences: A357482 A357483 A357484 * A357486 A357487 A357488

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 01 2022

STATUS

approved