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A325663 Matula-Goebel numbers of not necessarily regular rooted stars.

%I A325663 #11 Jul 26 2024 08:12:06
%S A325663 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,
%T A325663 44,45,48,50,54,55,60,62,64,66,72,75,80,81,88,90,93,96,99,100,108,110,
%U A325663 120,121,124,125,127,128,132,135,144,150,155,160,162,165,176
%N A325663 Matula-Goebel numbers of not necessarily regular rooted stars.
%C A325663 Products of members of A007097.
%C A325663 A rooted star is a rooted tree whose branches are all rooted paths.
%H A325663 Amiram Eldar, <a href="/A325663/b325663.txt">Table of n, a(n) for n = 1..10538</a> (terms up to A007097(12))
%F A325663 Sum_{n>=1} 1/a(n) = Product_{k>=1} A007097(k)/(A007097(k)-1) = 4.30328607286382284593... . - _Amiram Eldar_, Jul 26 2024
%e A325663 The sequence of rooted stars together with their Matula-Goebel numbers begins:
%e A325663    1: o
%e A325663    2: (o)
%e A325663    3: ((o))
%e A325663    4: (oo)
%e A325663    5: (((o)))
%e A325663    6: (o(o))
%e A325663    8: (ooo)
%e A325663    9: ((o)(o))
%e A325663   10: (o((o)))
%e A325663   11: ((((o))))
%e A325663   12: (oo(o))
%e A325663   15: ((o)((o)))
%e A325663   16: (oooo)
%e A325663   18: (o(o)(o))
%e A325663   20: (oo((o)))
%e A325663   22: (o(((o))))
%e A325663   24: (ooo(o))
%e A325663   25: (((o))((o)))
%e A325663   27: ((o)(o)(o))
%e A325663   30: (o(o)((o)))
%t A325663 rpQ[n_]:=n==1||PrimeQ[n]&&rpQ[PrimePi[n]];
%t A325663 Select[Range[100],And@@rpQ/@First/@FactorInteger[#]&]
%t A325663 (* 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 *)
%Y A325663 Cf. A007097, A056239, A061775, A109082, A109129, A112798, A196050, A324924, A325614, A325661, A325662.
%K A325663 nonn
%O A325663 1,2
%A A325663 _Gus Wiseman_, May 13 2019