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A316502 Matula-Goebel numbers of unlabeled rooted trees with n nodes in which the branches of any node with more than one branch have empty intersection.

%I A316502 #12 Aug 07 2018 07:11:32
%S A316502 1,2,3,4,5,6,7,8,10,11,12,13,14,15,16,17,18,19,20,22,24,26,28,29,30,
%T A316502 31,32,33,34,35,36,37,38,40,41,42,43,44,45,47,48,50,51,52,53,54,55,56,
%U A316502 58,59,60,61,62,64,66,67,68,70,71,72,74,75,76,77,78,79,80
%N A316502 Matula-Goebel numbers of unlabeled rooted trees with n nodes in which the branches of any node with more than one branch have empty intersection.
%C A316502 A prime index of n is a number m such that prime(m) divides n. A number is in the sequence iff it is 1, or either it is a prime or its prime indices are relatively prime, and its prime indices already belong to the sequence.
%H A316502 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Göbel numbers</a>
%e A316502 Sequence of rooted trees preceded by their Matula-Goebel numbers begins:
%e A316502    1: o
%e A316502    2: (o)
%e A316502    3: ((o))
%e A316502    4: (oo)
%e A316502    5: (((o)))
%e A316502    6: (o(o))
%e A316502    7: ((oo))
%e A316502    8: (ooo)
%e A316502   10: (o((o)))
%e A316502   11: ((((o))))
%e A316502   12: (oo(o))
%e A316502   13: ((o(o)))
%e A316502   14: (o(oo))
%e A316502   15: ((o)((o)))
%e A316502   16: (oooo)
%t A316502 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A316502 go[n_]:=Or[n==1,If[PrimeQ[n],go[PrimePi[n]],And[GCD@@primeMS[n]==1,And@@go/@primeMS[n]]]]
%t A316502 Select[Range[100],go]
%Y A316502 Cf. A000081, A000837, A007097, A276625, A289509, A302796, A316468, A316470, A316495, A316501, A316502.
%K A316502 nonn
%O A316502 1,2
%A A316502 _Gus Wiseman_, Jul 05 2018