cp's OEIS Frontend

A356237 Heinz numbers of integer partitions with a neighborless singleton.

%I A356237 #11 Aug 26 2022 23:40:50
%S A356237 2,3,5,7,10,11,13,14,17,19,20,21,22,23,26,28,29,31,33,34,37,38,39,40,
%T A356237 41,42,43,44,46,47,50,51,52,53,55,56,57,58,59,61,62,63,65,66,67,68,69,
%U A356237 70,71,73,74,76,78,79,80,82,83,84,85,86,87,88,89,91,92,93
%N A356237 Heinz numbers of integer partitions with a neighborless singleton.
%C A356237 A part x is neighborless if neither x - 1 nor x + 1 are parts, and a singleton if it appears only once.
%C A356237 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.
%C A356237 Also numbers that, for some prime index x, are not divisible by prime(x)^2, prime(x - 1), or prime(x + 1). Here, a prime index of n is a number m such that prime(m) divides n.
%e A356237 The terms together with their prime indices begin:
%e A356237    2: {1}
%e A356237    3: {2}
%e A356237    5: {3}
%e A356237    7: {4}
%e A356237   10: {1,3}
%e A356237   11: {5}
%e A356237   13: {6}
%e A356237   14: {1,4}
%e A356237   17: {7}
%e A356237   19: {8}
%e A356237   20: {1,1,3}
%e A356237   21: {2,4}
%e A356237   22: {1,5}
%e A356237   23: {9}
%e A356237   26: {1,6}
%e A356237   28: {1,1,4}
%t A356237 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356237 Select[Range[100],Function[ptn,Or@@Table[Count[ptn,x]==1&&!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]@*primeMS]
%Y A356237 The complement is counted by A355393.
%Y A356237 These partitions are counted by A356235.
%Y A356237 Not requiring a singleton gives A356734.
%Y A356237 A001221 counts distinct prime factors, with sum A001414.
%Y A356237 A003963 multiplies together the prime indices of n.
%Y A356237 A007690 counts partitions with no singletons, complement A183558.
%Y A356237 A056239 adds up prime indices, row sums of A112798, lengths A001222.
%Y A356237 A073491 lists numbers with gapless prime indices, complement A073492.
%Y A356237 A132747 counts non-isolated divisors, complement A132881.
%Y A356237 A356069 counts gapless divisors, initial A356224 (complement A356225).
%Y A356237 A356236 counts partitions with a neighborless part, complement A355394.
%Y A356237 A356607 counts strict partitions w/ a neighborless part, complement A356606.
%Y A356237 Cf. A286470, A289508, A325160, A328166, A328335, A356231, A356233, A356234.
%K A356237 nonn
%O A356237 1,1
%A A356237 _Gus Wiseman_, Aug 24 2022