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A325662
Matula-Goebel numbers of regular rooted stars.
3
1, 2, 3, 4, 5, 8, 9, 11, 16, 25, 27, 31, 32, 64, 81, 121, 125, 127, 128, 243, 256, 512, 625, 709, 729, 961, 1024, 1331, 2048, 2187, 3125, 4096, 5381, 6561, 8192, 14641, 15625, 16129, 16384, 19683, 29791, 32768, 52711, 59049, 65536, 78125, 131072, 161051
OFFSET
1,2
COMMENTS
Powers of members of A007097.
A regular rooted star is a rooted tree whose branches are all rooted paths of equal length.
The number of terms <= 10^k, k=0,1,2,...: 1, 7, 15, 26, 35, 46, 56, 67, 76, 87, 98, 109, 121, 131, 142, 154, 163, 175, 185, 198, 208, 220, 231, 241, 254, 265, 275, etc. - Robert G. Wilson v, May 13 2019
LINKS
Robert G. Wilson v, Table of n, a(n) for n = 1..275 (terms 1..48 from Gus Wiseman)
FORMULA
Sum_{n>=1} 1/a(n) = 1 + Product_{k>=1} 1/(A007097(k)-1) = 2.8928887669834086909... - Amiram Eldar, Jul 26 2024
EXAMPLE
The sequence of regular rooted stars together with their Matula-Goebel numbers begins:
1: o
2: (o)
3: ((o))
4: (oo)
5: (((o)))
8: (ooo)
9: ((o)(o))
11: ((((o))))
16: (oooo)
25: (((o))((o)))
27: ((o)(o)(o))
31: (((((o)))))
32: (ooooo)
64: (oooooo)
81: ((o)(o)(o)(o))
121: ((((o)))(((o))))
125: (((o))((o))((o)))
127: ((((((o))))))
128: (ooooooo)
MATHEMATICA
rpQ[n_]:=n==1||PrimeQ[n]&&rpQ[PrimePi[n]];
Select[Range[100], #==1||PrimePowerQ[#]&&rpQ[FactorInteger[#][[1, 1]]]&]
(* generates terms <= A007097(max) *) seq[max_] := Module[{ps = NestList[Prime@# &, 1, max], psmax, s = {1}, emax, s1}, pmax = Max[ps]; Do[p = ps[[k]]; emax = Floor[Log[p, pmax]]; s1 = p^Range[emax]; s = Union[s, s1], {k, 2, Length[ps]}]; s]; seq[10] (* Amiram Eldar, Jul 26 2024 *)
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 13 2019
STATUS
approved