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A318690
Matula-Goebel numbers of powerful uniform rooted trees.
3
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.
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]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 31 2018
STATUS
approved