A316467 Matula-Goebel numbers of locally stable rooted identity trees, meaning no branch is a subset of any other branch of the same root.

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

Author

Gus Wiseman, Jul 04 2018

Keywords

• nonn

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.

Examples

			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))))
		

Crossrefs

Cf. A000081, A004111, A007097, A276625, A277098, A285572, A285573, A302796, A303362, A304713, A316468, A316469, A316471, A316474, A316476, A316494.

Programs

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]