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A356736 Heinz numbers of integer partitions with no neighborless parts.

%I A356736 #10 Sep 01 2022 09:33:46
%S A356736 1,6,12,15,18,24,30,35,36,45,48,54,60,72,75,77,90,96,105,108,120,135,
%T A356736 143,144,150,162,175,180,192,210,216,221,225,240,245,270,288,300,315,
%U A356736 323,324,360,375,384,385,405,420,432,437,450,462,480,486,525,539,540
%N A356736 Heinz numbers of integer partitions with no neighborless parts.
%C A356736 First differs from A066312 in having 1 and lacking 462.
%C A356736 First differs from A104210 in having 1 and lacking 42.
%C A356736 A part x is neighborless iff neither x - 1 nor x + 1 are parts.
%C A356736 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%e A356736 The terms together with their prime indices begin:
%e A356736    1: {}
%e A356736    6: {1,2}
%e A356736   12: {1,1,2}
%e A356736   15: {2,3}
%e A356736   18: {1,2,2}
%e A356736   24: {1,1,1,2}
%e A356736   30: {1,2,3}
%e A356736   35: {3,4}
%e A356736   36: {1,1,2,2}
%e A356736   45: {2,2,3}
%e A356736   48: {1,1,1,1,2}
%e A356736   54: {1,2,2,2}
%e A356736   60: {1,1,2,3}
%e A356736   72: {1,1,1,2,2}
%e A356736   75: {2,3,3}
%e A356736   77: {4,5}
%e A356736   90: {1,2,2,3}
%e A356736   96: {1,1,1,1,1,2}
%t A356736 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356736 Select[Range[100],Function[ptn,!Or@@Table[!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]@*primeMS]
%Y A356736 These partitions are counted by A355394.
%Y A356736 The singleton case is the complement of A356237.
%Y A356736 The singleton case is counted by A355393, complement A356235.
%Y A356736 The strict complement is A356606, counted by A356607.
%Y A356736 The complement is A356734, counted by A356236.
%Y A356736 A000041 counts integer partitions, strict A000009.
%Y A356736 A001221 counts distinct prime factors, sum A001414.
%Y A356736 A003963 multiplies together the prime indices of n.
%Y A356736 A007690 counts partitions with no singletons, complement A183558.
%Y A356736 A056239 adds up prime indices, row sums of A112798, lengths A001222.
%Y A356736 A073491 lists numbers with gapless prime indices, complement A073492.
%Y A356736 Cf. A066312, A286470, A287170 (firsts A066205), A328171, A328187, A328221, A328335, A356231, A356234.
%K A356736 nonn
%O A356736 1,2
%A A356736 _Gus Wiseman_, Aug 31 2022