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

1,2

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.

Links

Table of n, a(n) for n=1..59.

Gus Wiseman, The first 96 powerful uniform rooted trees.

Example

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))

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]

Crossrefs

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

Sequence in context: A184155 A331913 A318612 * A302498 A243497 A214577

Adjacent sequences: A318687 A318688 A318689 * A318691 A318692 A318693

Keyword

nonn

Author

Gus Wiseman, Aug 31 2018

Status

approved