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.
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 04 2018
EXTENSIONS
Rewritten by Gus Wiseman, Aug 31 2018
STATUS
approved