A356237 Heinz numbers of integer partitions with a neighborless singleton.

2, 3, 5, 7, 10, 11, 13, 14, 17, 19, 20, 21, 22, 23, 26, 28, 29, 31, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93

Offset: 1

Author

Gus Wiseman, Aug 24 2022

Keywords

• nonn

Comments

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

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.

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.

Examples

			The terms together with their prime indices begin:
   2: {1}
   3: {2}
   5: {3}
   7: {4}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}
  26: {1,6}
  28: {1,1,4}
		

Crossrefs

The complement is counted by A355393.

These partitions are counted by A356235.

Not requiring a singleton gives A356734.

A001221 counts distinct prime factors, with sum A001414.

A003963 multiplies together the prime indices of n.

A007690 counts partitions with no singletons, complement A183558.

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

A073491 lists numbers with gapless prime indices, complement A073492.

A132747 counts non-isolated divisors, complement A132881.

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

A356236 counts partitions with a neighborless part, complement A355394.

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

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

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Function[ptn,Or@@Table[Count[ptn,x]==1&&!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]@*primeMS]