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A325663
Matula-Goebel numbers of not necessarily regular rooted stars.
3
1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 18, 20, 22, 24, 25, 27, 30, 31, 32, 33, 36, 40, 44, 45, 48, 50, 54, 55, 60, 62, 64, 66, 72, 75, 80, 81, 88, 90, 93, 96, 99, 100, 108, 110, 120, 121, 124, 125, 127, 128, 132, 135, 144, 150, 155, 160, 162, 165, 176
OFFSET
1,2
COMMENTS
Products of members of A007097.
A rooted star is a rooted tree whose branches are all rooted paths.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10538 (terms up to A007097(12))
FORMULA
Sum_{n>=1} 1/a(n) = Product_{k>=1} A007097(k)/(A007097(k)-1) = 4.30328607286382284593... . - Amiram Eldar, Jul 26 2024
EXAMPLE
The sequence of rooted stars together with their Matula-Goebel numbers begins:
1: o
2: (o)
3: ((o))
4: (oo)
5: (((o)))
6: (o(o))
8: (ooo)
9: ((o)(o))
10: (o((o)))
11: ((((o))))
12: (oo(o))
15: ((o)((o)))
16: (oooo)
18: (o(o)(o))
20: (oo((o)))
22: (o(((o))))
24: (ooo(o))
25: (((o))((o)))
27: ((o)(o)(o))
30: (o(o)((o)))
MATHEMATICA
rpQ[n_]:=n==1||PrimeQ[n]&&rpQ[PrimePi[n]];
Select[Range[100], And@@rpQ/@First/@FactorInteger[#]&]
(* generates terms <= A007097(max) *) seq[max_] := Module[{ps = NestList[Prime@# &, 1, max], psmax, s = {1}, emax, s1, s2}, pmax = Max[ps]; Do[p = ps[[k]]; emax = Floor[Log[p, pmax]]; s1 = p^Range[0, emax]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= pmax &]; s = Union[s, s2], {k, 2, Length[ps]}]; s]; seq[7] (* Amiram Eldar, Jul 26 2024 *)
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 13 2019
STATUS
approved