OFFSET
1,1
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.
EXAMPLE
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}
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 24 2022
STATUS
approved