OFFSET
1,2
COMMENTS
Every positive integer has a unique factorization into factors q(i) = prime(i)/i, i > 0. This sequence consists of all numbers where this factorization has all distinct factors, except possibly for any multiplicity of q(1). For example, 22 = q(1)^2 q(2) q(3) q(5) is in the sequence, while 50 = q(1)^3 q(2)^2 q(3)^2 is not.
The enumeration of these trees by number of vertices is A324936.
LINKS
EXAMPLE
The sequence of trees together with their Matula-Goebel numbers begins:
1: o
2: (o)
3: ((o))
4: (oo)
5: (((o)))
6: (o(o))
7: ((oo))
8: (ooo)
10: (o((o)))
11: ((((o))))
12: (oo(o))
13: ((o(o)))
14: (o(oo))
16: (oooo)
17: (((oo)))
19: ((ooo))
20: (oo((o)))
21: ((o)(oo))
22: (o(((o))))
24: (ooo(o))
26: (o(o(o)))
28: (oo(oo))
29: ((o((o))))
31: (((((o)))))
MATHEMATICA
difac[n_]:=If[n==1, {}, With[{i=PrimePi[FactorInteger[n][[1, 1]]]}, Sort[Prepend[difac[n*i/Prime[i]], i]]]];
Select[Range[100], UnsameQ@@DeleteCases[difac[#], 1]&]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 21 2019
STATUS
approved