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A316794
Matula-Goebel numbers of aperiodic rooted trees with locally distinct multiplicities.
4
1, 2, 3, 5, 11, 12, 18, 20, 24, 31, 37, 40, 44, 45, 48, 50, 54, 61, 71, 72, 75, 80, 88, 89, 96, 99, 108, 124, 127, 135, 148, 157, 160, 162, 173, 176, 192, 193, 197, 200, 223, 229, 242, 244, 248, 250, 251, 275, 279, 283, 284, 288, 296, 297, 320, 333, 352, 353
OFFSET
1,2
COMMENTS
A positive integer belongs to the sequence iff either it is equal to 1 or it belongs to A007916 (numbers that are not perfect powers, or numbers whose prime multiplicities are relatively prime) as well as to A130091 (numbers whose prime multiplicities are distinct), and all of its prime indices already belong to the sequence. A prime index of n is a number m such that prime(m) divides n.
EXAMPLE
Sequence of aperiodic rooted trees with locally distinct multiplicities preceded by their Matula-Goebel numbers begins:
1: o
2: (o)
3: ((o))
5: (((o)))
11: ((((o))))
12: (oo(o))
18: (o(o)(o))
20: (oo((o)))
24: (ooo(o))
31: (((((o)))))
37: ((oo(o)))
40: (ooo((o)))
44: (oo(((o))))
45: ((o)(o)((o)))
48: (oooo(o))
50: (o((o))((o)))
MATHEMATICA
mgsbQ[n_]:=Or[n==1, And[UnsameQ@@Last/@FactorInteger[n], GCD@@Last/@FactorInteger[n]==1, And@@Cases[FactorInteger[n], {p_, _}:>mgsbQ[PrimePi[p]]]]];
Select[Range[100], mgsbQ]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 14 2018
STATUS
approved