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%I #14 May 15 2019 04:58:12
%S 1,2,3,4,5,8,9,11,16,25,27,31,32,64,81,121,125,127,128,243,256,512,
%T 625,709,729,961,1024,1331,2048,2187,3125,4096,5381,6561,8192,14641,
%U 15625,16129,16384,19683,29791,32768,52711,59049,65536,78125,131072,161051
%N Matula-Goebel numbers of regular rooted stars.
%C Powers of members of A007097.
%C A regular rooted star is a rooted tree whose branches are all rooted paths of equal length.
%C 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 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_)
%e The sequence of regular rooted stars together with their Matula-Goebel numbers begins:
%e 1: o
%e 2: (o)
%e 3: ((o))
%e 4: (oo)
%e 5: (((o)))
%e 8: (ooo)
%e 9: ((o)(o))
%e 11: ((((o))))
%e 16: (oooo)
%e 25: (((o))((o)))
%e 27: ((o)(o)(o))
%e 31: (((((o)))))
%e 32: (ooooo)
%e 64: (oooooo)
%e 81: ((o)(o)(o)(o))
%e 121: ((((o)))(((o))))
%e 125: (((o))((o))((o)))
%e 127: ((((((o))))))
%e 128: (ooooooo)
%t rpQ[n_]:=n==1||PrimeQ[n]&&rpQ[PrimePi[n]];
%t Select[Range[100],#==1||PrimePowerQ[#]&&rpQ[FactorInteger[#][[1,1]]]&]
%Y Cf. A007097, A056239, A061775, A109082, A109129, A112798, A196050, A324924, A325614, A325661, A325663.
%K nonn
%O 1,2
%A _Gus Wiseman_, May 13 2019
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