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A317705
Matula-Goebel numbers of series-reduced powerful rooted trees.
15
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.
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 *)
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 04 2018
EXTENSIONS
Rewritten by Gus Wiseman, Aug 31 2018
STATUS
approved