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

1,2

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.

Links

Table of n, a(n) for n=1..58.

Example

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)))

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]

Crossrefs

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

Sequence in context: A210144 A243357 A066159 * A103027 A093902 A269004

Adjacent sequences: A316791 A316792 A316793 * A316795 A316796 A316797

Keyword

nonn

Author

Gus Wiseman, Jul 14 2018

Status

approved