A356237 - OEIS

%I #11 Aug 26 2022 23:40:50

%S 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 41,42,43,44,46,47,50,51,52,53,55,56,57,58,59,61,62,63,65,66,67,68,69,

%U 70,71,73,74,76,78,79,80,82,83,84,85,86,87,88,89,91,92,93

%N Heinz numbers of integer partitions with a neighborless singleton.

%C A part x is neighborless if neither x - 1 nor x + 1 are parts, and a singleton if it appears only once.

%C 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 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 The terms together with their prime indices begin:

%e    2: {1}

%e    3: {2}

%e    5: {3}

%e    7: {4}

%e   10: {1,3}

%e   11: {5}

%e   13: {6}

%e   14: {1,4}

%e   17: {7}

%e   19: {8}

%e   20: {1,1,3}

%e   21: {2,4}

%e   22: {1,5}

%e   23: {9}

%e   26: {1,6}

%e   28: {1,1,4}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],Function[ptn,Or@@Table[Count[ptn,x]==1&&!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]@*primeMS]

%Y The complement is counted by A355393.

%Y These partitions are counted by A356235.

%Y Not requiring a singleton gives A356734.

%Y A001221 counts distinct prime factors, with sum A001414.

%Y A003963 multiplies together the prime indices of n.

%Y A007690 counts partitions with no singletons, complement A183558.

%Y A056239 adds up prime indices, row sums of A112798, lengths A001222.

%Y A073491 lists numbers with gapless prime indices, complement A073492.

%Y A132747 counts non-isolated divisors, complement A132881.

%Y A356069 counts gapless divisors, initial A356224 (complement A356225).

%Y A356236 counts partitions with a neighborless part, complement A355394.

%Y A356607 counts strict partitions w/ a neighborless part, complement A356606.

%Y Cf. A286470, A289508, A325160, A328166, A328335, A356231, A356233, A356234.

%K nonn

%O 1,1

%A _Gus Wiseman_, Aug 24 2022