A317705 Matula-Goebel numbers of series-reduced powerful rooted trees.

1, 4, 8, 16, 32, 49, 64, 128, 196, 256, 343, 361, 392, 512, 784, 1024, 1372, 1444, 1568, 2048, 2401, 2744, 2809, 2888, 3136, 4096, 5488, 5776, 6272, 6859, 8192, 9604, 10976, 11236, 11552, 12544, 16384, 16807, 17161, 17689, 19208, 21952, 22472, 23104, 25088

Offset

1,2

Comments

A positive integer n is a Matula-Goebel number of a series-reduced powerful rooted tree iff either n = 1 or n is a powerful number (meaning its prime multiplicities are all greater than 1) whose prime indices are all Matula-Goebel numbers of series-reduced powerful rooted trees, where a prime index of n is a number m such that prime(m) divides n.

Links

Table of n, a(n) for n=1..45.

Index entries for sequences related to Matula-Göbel numbers

Example

The sequence of Matula-Goebel numbers of series-reduced powerful rooted trees together with the corresponding trees begins:
    1: o
    4: (oo)
    8: (ooo)
   16: (oooo)
   32: (ooooo)
   49: ((oo)(oo))
   64: (oooooo)
  128: (ooooooo)
  196: (oo(oo)(oo))
  256: (oooooooo)
  343: ((oo)(oo)(oo))
  361: ((ooo)(ooo))
  392: (ooo(oo)(oo))
  512: (ooooooooo)
  784: (oooo(oo)(oo))

Mathematica

powgoQ[n_]:=Or[n==1, And[Min@@FactorInteger[n][[All, 2]]>1, And@@powgoQ/@PrimePi/@FactorInteger[n][[All, 1]]]];
Select[Range[1000], powgoQ] (* Gus Wiseman, Aug 31 2018 *)
(* Second program: *)
Nest[Function[a, Append[a, Block[{k = a[[-1]] + 1}, While[Nand[AllTrue[#[[All, -1]], # > 1 & ], AllTrue[PrimePi[#[[All, 1]] ], MemberQ[a, #] &]] &@ FactorInteger@ k, k++]; k]]], {1}, 44] (* Michael De Vlieger, Aug 05 2018 *)

Crossrefs

Cf. A000081, A001694, A061775, A111299, A214577, A276625, A277098, A303431.

Cf. A317102, A317707, A317708, A317709, A317710, A317711, A317712, A317717, A317718, A317719.

Sequence in context: A293780 A048168 A175341 * A318692 A291441 A331967

Adjacent sequences: A317702 A317703 A317704 * A317706 A317707 A317708

Keyword

nonn

Author

Gus Wiseman, Aug 04 2018

Extensions

Rewritten by Gus Wiseman, Aug 31 2018

Status

approved