A318690 Matula-Goebel numbers of powerful uniform rooted trees.

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 23, 25, 27, 31, 32, 36, 49, 53, 59, 64, 67, 81, 83, 97, 100, 103, 121, 125, 127, 128, 131, 151, 196, 216, 225, 227, 241, 243, 256, 277, 289, 311, 331, 343, 361, 419, 431, 441, 484, 509, 512, 529, 541, 563, 625, 661, 691

Offset: 1

Author

Gus Wiseman, Aug 31 2018

Keywords

• nonn

Comments

A prime index of n is a number m such that prime(m) divides n. A positive integer n is a Matula-Goebel number of a powerful uniform rooted tree iff either n = 1 or n is a prime number whose prime index is a Matula-Goebel number of a powerful uniform rooted tree or n is a squarefree number taken to a power > 1 whose prime indices are all Matula-Goebel numbers of powerful uniform rooted trees.

Examples

			The sequence of all powerful uniform rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   7: ((oo))
   8: (ooo)
   9: ((o)(o))
  11: ((((o))))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  31: (((((o)))))
  32: (ooooo)
  36: (oo(o)(o))
  49: ((oo)(oo))
		

Links

• Gus Wiseman, The first 96 powerful uniform rooted trees.

Crossrefs

Cf. A000081, A001694, A061775, A072774, A214577, A317705, A317707, A317710, A317717, A317719, A318611, A318612, A318689, A318692.

Programs

Mathematica

powunQ[n_]:=Or[n==1,If[PrimeQ[n],powunQ[PrimePi[n]],And[SameQ@@FactorInteger[n][[All,2]],Min@@FactorInteger[n][[All,2]]>1,And@@powunQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[100],powunQ]