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%I #6 Jan 27 2022 20:46:31
%S 20,28,40,44,52,56,63,68,76,80,84,88,92,99,100,104,112,116,117,124,
%T 126,132,136,140,148,152,153,156,160,164,168,171,172,176,184,188,189,
%U 196,198,200,204,207,208,212,220,224,228,232,234,236,244,248,252,260,261
%N Heinz numbers of integer partitions with a difference < -1 and a conjugate difference < -1.
%C We define a difference of a partition to be a difference of two adjacent parts.
%e The terms together with their prime indices begin:
%e 20: (3,1,1)
%e 28: (4,1,1)
%e 40: (3,1,1,1)
%e 44: (5,1,1)
%e 52: (6,1,1)
%e 56: (4,1,1,1)
%e 63: (4,2,2)
%e 68: (7,1,1)
%e 76: (8,1,1)
%e 80: (3,1,1,1,1)
%e 84: (4,2,1,1)
%e 88: (5,1,1,1)
%e 92: (9,1,1)
%e 99: (5,2,2)
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t Select[Range[100],(Min@@Differences[Reverse[primeMS[#]]]<-1)&&(Min@@Differences[conj[primeMS[#]]]<-1)&]
%Y Heinz number rankings are in parentheses below.
%Y Taking just one condition gives (A073492) and (A065201), counted by A239955.
%Y These partitions are counted by A350839.
%Y A000041 = integer partitions, strict A000009.
%Y A034296 = partitions with no gaps (A073491), strict A001227 (A073485).
%Y A090858 = partitions with a single gap of size 1 (A325284).
%Y A116931 = partitions with no successions (A319630), strict A003114.
%Y A116932 = partitions with no successions or gaps of size 1, strict A025157.
%Y A350842 = partitions with no gaps of size 1, strict A350844, sets A005314.
%Y Cf. A000070, A024619, A026424, A055932, A183558, A277103, A305148, A321440, A350838.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jan 26 2022
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