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A350841
Heinz numbers of integer partitions with a difference < -1 and a conjugate difference < -1.
12
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
1,1
COMMENTS
We define a difference of a partition to be a difference of two adjacent parts.
EXAMPLE
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)
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)&]
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.
Sequence in context: A137428 A368089 A367455 * A361867 A309769 A252478
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 26 2022
STATUS
approved