A317493 Heinz numbers of integer partitions that are not fully normal.

9, 24, 25, 27, 36, 40, 48, 49, 54, 56, 72, 80, 81, 88, 96, 100, 104, 108, 112, 120, 121, 125, 135, 136, 144, 152, 160, 162, 168, 169, 176, 184, 189, 192, 196, 200, 208, 216, 224, 225, 232, 240, 243, 248, 250, 264, 270, 272, 280, 288, 289, 296, 297, 304, 312

Offset

1,1

Comments

An integer partition is fully normal if either it is of the form (1,1,...,1) or its multiplicities span an initial interval of positive integers and, sorted in weakly decreasing order, are themselves fully normal.

Links

Table of n, a(n) for n=1..55.

Example

Sequence of all integer partitions that are not fully normal begins: (22), (2111), (33), (222), (2211), (3111), (21111), (44), (2221), (4111), (22111), (31111), (2222), (5111), (211111), (3311).

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
fulnrmQ[ptn_]:=With[{qtn=Sort[Length/@Split[ptn], Greater]}, Or[ptn=={}||Union[ptn]=={1}, And[Union[qtn]==Range[Max[qtn]], fulnrmQ[qtn]]]];
Select[Range[100], !fulnrmQ[Reverse[primeMS[#]]]&]

Crossrefs

Cf. A055932, A056239, A181819, A182850, A296150, A305733, A317089, A317090, A317245, A317246, A317491, A317492.

Sequence in context: A161431 A223379 A223297 * A102217 A124065 A121453

Adjacent sequences: A317490 A317491 A317492 * A317494 A317495 A317496

Keyword

nonn

Author

Gus Wiseman, Jul 30 2018

Status

approved