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A325662 Matula-Goebel numbers of regular rooted stars.

%I A325662 #20 Jul 26 2024 08:57:39
%S A325662 1,2,3,4,5,8,9,11,16,25,27,31,32,64,81,121,125,127,128,243,256,512,
%T A325662 625,709,729,961,1024,1331,2048,2187,3125,4096,5381,6561,8192,14641,
%U A325662 15625,16129,16384,19683,29791,32768,52711,59049,65536,78125,131072,161051
%N A325662 Matula-Goebel numbers of regular rooted stars.
%C A325662 Powers of members of A007097.
%C A325662 A regular rooted star is a rooted tree whose branches are all rooted paths of equal length.
%C A325662 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
%H A325662 Robert G. Wilson v, <a href="/A325662/b325662.txt">Table of n, a(n) for n = 1..275</a> (terms 1..48 from _Gus Wiseman_)
%F A325662 Sum_{n>=1} 1/a(n) = 1 + Product_{k>=1} 1/(A007097(k)-1) = 2.8928887669834086909... - _Amiram Eldar_, Jul 26 2024
%e A325662 The sequence of regular rooted stars together with their Matula-Goebel numbers begins:
%e A325662     1: o
%e A325662     2: (o)
%e A325662     3: ((o))
%e A325662     4: (oo)
%e A325662     5: (((o)))
%e A325662     8: (ooo)
%e A325662     9: ((o)(o))
%e A325662    11: ((((o))))
%e A325662    16: (oooo)
%e A325662    25: (((o))((o)))
%e A325662    27: ((o)(o)(o))
%e A325662    31: (((((o)))))
%e A325662    32: (ooooo)
%e A325662    64: (oooooo)
%e A325662    81: ((o)(o)(o)(o))
%e A325662   121: ((((o)))(((o))))
%e A325662   125: (((o))((o))((o)))
%e A325662   127: ((((((o))))))
%e A325662   128: (ooooooo)
%t A325662 rpQ[n_]:=n==1||PrimeQ[n]&&rpQ[PrimePi[n]];
%t A325662 Select[Range[100],#==1||PrimePowerQ[#]&&rpQ[FactorInteger[#][[1,1]]]&]
%t A325662 (* 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 *)
%Y A325662 Cf. A007097, A056239, A061775, A109082, A109129, A112798, A196050, A324924, A325614, A325661, A325663.
%K A325662 nonn
%O A325662 1,2
%A A325662 _Gus Wiseman_, May 13 2019