A317258 Heinz numbers of integer partitions that are not totally nonincreasing.

18, 50, 54, 75, 90, 98, 108, 126, 147, 150, 162, 180, 198, 234, 242, 245, 250, 252, 270, 294, 300, 306, 324, 338, 342, 350, 363, 375, 378, 396, 414, 450, 468, 486, 490, 500, 507, 522, 525, 540, 550, 558, 578, 588, 594, 600, 605, 612, 630, 648, 650, 666, 684

Offset: 1

Author

Gus Wiseman, Jul 25 2018

Keywords

• nonn

Comments

An integer partition is totally nonincreasing if either it is empty or a singleton or its multiplicities (where if x < y the multiplicity of x is counted prior to the multiplicity of y) are weakly decreasing and are themselves a totally nonincreasing integer partition.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			Sequence of all integer partitions that are not totally nonincreasing begins: (221), (331), (2221), (332), (3221), (441), (22211), (4221), (442), (3321), (22221), (32211), (5221), (6221), (551), (443), (3331), (42211), (32221), (4421), (33211), (7221), (222211), (661), (8221), (4331), (552), (3332), (42221), (52211), (9221), (33221).
		

Crossrefs

Cf. A056239, A071365, A100883, A112769, A181819, A182850, A242031, A296150, A305733, A317256, A317257.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
totincQ[q_]:=Or[Length[q]<=1,And[OrderedQ[Length/@Split[q]],totincQ[Reverse[Length/@Split[q]]]]];
Select[Range[1000],!totincQ[Reverse[primeMS[#]]]&]