A316794 Matula-Goebel numbers of aperiodic rooted trees with locally distinct multiplicities.

1, 2, 3, 5, 11, 12, 18, 20, 24, 31, 37, 40, 44, 45, 48, 50, 54, 61, 71, 72, 75, 80, 88, 89, 96, 99, 108, 124, 127, 135, 148, 157, 160, 162, 173, 176, 192, 193, 197, 200, 223, 229, 242, 244, 248, 250, 251, 275, 279, 283, 284, 288, 296, 297, 320, 333, 352, 353

Offset: 1

Author

Gus Wiseman, Jul 14 2018

Keywords

• nonn

Comments

A positive integer belongs to the sequence iff either it is equal to 1 or it belongs to A007916 (numbers that are not perfect powers, or numbers whose prime multiplicities are relatively prime) as well as to A130091 (numbers whose prime multiplicities are distinct), and all of its prime indices already belong to the sequence. A prime index of n is a number m such that prime(m) divides n.

Examples

			Sequence of aperiodic rooted trees with locally distinct multiplicities preceded by their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   5: (((o)))
  11: ((((o))))
  12: (oo(o))
  18: (o(o)(o))
  20: (oo((o)))
  24: (ooo(o))
  31: (((((o)))))
  37: ((oo(o)))
  40: (ooo((o)))
  44: (oo(((o))))
  45: ((o)(o)((o)))
  48: (oooo(o))
  50: (o((o))((o)))
		

Crossrefs

Cf. A000081, A004111, A007097, A007916, A061775, A276625, A301700 A303431, A316793, A316795, A316796.

Programs

Mathematica

mgsbQ[n_]:=Or[n==1,And[UnsameQ@@Last/@FactorInteger[n],GCD@@Last/@FactorInteger[n]==1,And@@Cases[FactorInteger[n],{p_,_}:>mgsbQ[PrimePi[p]]]]];
Select[Range[100],mgsbQ]