A350841 Heinz numbers of integer partitions with a difference < -1 and a conjugate difference < -1.

20, 28, 40, 44, 52, 56, 63, 68, 76, 80, 84, 88, 92, 99, 100, 104, 112, 116, 117, 124, 126, 132, 136, 140, 148, 152, 153, 156, 160, 164, 168, 171, 172, 176, 184, 188, 189, 196, 198, 200, 204, 207, 208, 212, 220, 224, 228, 232, 234, 236, 244, 248, 252, 260, 261

Offset: 1

Author

Gus Wiseman, Jan 26 2022

Keywords

• nonn

Comments

We define a difference of a partition to be a difference of two adjacent parts.

Examples

			The terms together with their prime indices begin:
   20: (3,1,1)
   28: (4,1,1)
   40: (3,1,1,1)
   44: (5,1,1)
   52: (6,1,1)
   56: (4,1,1,1)
   63: (4,2,2)
   68: (7,1,1)
   76: (8,1,1)
   80: (3,1,1,1,1)
   84: (4,2,1,1)
   88: (5,1,1,1)
   92: (9,1,1)
   99: (5,2,2)
		

Crossrefs

Heinz number rankings are in parentheses below.

Taking just one condition gives (A073492) and (A065201), counted by A239955.

These partitions are counted by A350839.

A000041 = integer partitions, strict A000009.

A034296 = partitions with no gaps (A073491), strict A001227 (A073485).

A090858 = partitions with a single gap of size 1 (A325284).

A116931 = partitions with no successions (A319630), strict A003114.

A116932 = partitions with no successions or gaps of size 1, strict A025157.

A350842 = partitions with no gaps of size 1, strict A350844, sets A005314.

Cf. A000070, A024619, A026424, A055932, A183558, A277103, A305148, A321440, A350838.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],(Min@@Differences[Reverse[primeMS[#]]]<-1)&&(Min@@Differences[conj[primeMS[#]]]<-1)&]