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A316467
Matula-Goebel numbers of locally stable rooted identity trees, meaning no branch is a subset of any other branch of the same root.
9
1, 2, 3, 5, 11, 15, 31, 33, 47, 55, 93, 127, 137, 141, 155, 165, 211, 257, 341, 381, 411, 465, 487, 633, 635, 709, 771, 773, 811, 907, 977, 1023, 1055, 1285, 1297, 1397, 1457, 1461, 1507, 1621, 1705, 1905, 2127, 2293, 2319, 2321, 2433, 2621, 2721, 2833, 2931
OFFSET
1,2
COMMENTS
A prime index of n is a number m such that prime(m) divides n. A number belongs to this sequence iff it is squarefree, its distinct prime indices are pairwise indivisible, and its prime indices also belong to this sequence.
EXAMPLE
165 = prime(2)*prime(3)*prime(5) belongs to the sequence because it is squarefree, the indices {2,3,5} are pairwise indivisible, and each of them already belongs to the sequence.
Sequence of locally stable rooted identity trees preceded by their Matula-Goebel numbers begins:
1: o
2: (o)
3: ((o))
5: (((o)))
11: ((((o))))
15: ((o)((o)))
31: (((((o)))))
33: ((o)(((o))))
47: (((o)((o))))
55: (((o))(((o))))
93: ((o)((((o)))))
127: ((((((o))))))
137: (((o)(((o)))))
141: ((o)((o)((o))))
155: (((o))((((o)))))
165: ((o)((o))(((o))))
MATHEMATICA
primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
ain[n_]:=And[Select[Tuples[primeMS[n], 2], UnsameQ@@#&&Divisible@@#&]=={}, SquareFreeQ[n], And@@ain/@primeMS[n]];
Select[Range[100], ain]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 04 2018
STATUS
approved