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A356734 Heinz numbers of integer partitions with at least one neighborless part.

%I A356734 #7 Aug 30 2022 09:41:50
%S A356734 2,3,4,5,7,8,9,10,11,13,14,16,17,19,20,21,22,23,25,26,27,28,29,31,32,
%T A356734 33,34,37,38,39,40,41,42,43,44,46,47,49,50,51,52,53,55,56,57,58,59,61,
%U A356734 62,63,64,65,66,67,68,69,70,71,73,74,76,78,79,80,81,82,83
%N A356734 Heinz numbers of integer partitions with at least one neighborless part.
%C A356734 First differs from A319630 in lacking 1 and having 42 (prime indices: {1,2,4}).
%C A356734 A part x is neighborless if neither x - 1 nor x + 1 are parts.
%C A356734 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 A356734 The terms together with their prime indices begin:
%e A356734     2: {1}
%e A356734     3: {2}
%e A356734     4: {1,1}
%e A356734     5: {3}
%e A356734     7: {4}
%e A356734     8: {1,1,1}
%e A356734     9: {2,2}
%e A356734    10: {1,3}
%e A356734    11: {5}
%e A356734    13: {6}
%e A356734    14: {1,4}
%e A356734    16: {1,1,1,1}
%e A356734    17: {7}
%e A356734    19: {8}
%e A356734    20: {1,1,3}
%e A356734    21: {2,4}
%e A356734    22: {1,5}
%t A356734 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356734 Select[Range[100],Function[ptn,Or@@Table[!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]@*primeMS]
%Y A356734 These partitions are counted by A356236.
%Y A356734 The singleton case is A356237, counted by A356235 (complement A355393).
%Y A356734 The strict case is counted by A356607, complement A356606.
%Y A356734 The complement is A356736, counted by A355394.
%Y A356734 A001221 counts distinct prime factors, sum A001414.
%Y A356734 A003963 multiplies together the prime indices of n.
%Y A356734 A007690 counts partitions with no singletons, complement A183558.
%Y A356734 A056239 adds up prime indices, row sums of A112798, lengths A001222.
%Y A356734 A073491 lists numbers with gapless prime indices, complement A073492.
%Y A356734 A132747 counts non-isolated divisors, complement A132881.
%Y A356734 A356069 counts gapless divisors, initial A356224 (complement A356225).
%Y A356734 Cf. A000005, A286470, A287170 (firsts A066205), A289508, A325160, A328166, A328335, A356231, A356233, A356234.
%K A356734 nonn
%O A356734 1,1
%A A356734 _Gus Wiseman_, Aug 26 2022