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A292127 a(1) = 1, a(r(n)^k) = 1 + k * a(n) where r(n) is the n-th number that is not a perfect power A007916(n).

%I A292127 #10 May 04 2018 22:42:29
%S A292127 1,2,3,3,4,4,5,4,5,5,6,5,6,6,7,5,6,7,7,8,6,7,8,8,7,9,7,7,8,9,9,6,8,10,
%T A292127 8,7,8,9,10,10,7,9,11,9,8,9,10,11,9,11,8,10,12,10,9,10,11,12,10,12,9,
%U A292127 11,13,7,11,10,11,12,13,11,13,10,12,14,8,12,11
%N A292127 a(1) = 1, a(r(n)^k) = 1 + k * a(n) where r(n) is the n-th number that is not a perfect power A007916(n).
%C A292127 Any positive integer greater than 1 can be written uniquely as a perfect power r(n)^k. We define a planted achiral (or generalized Bethe) tree b(n) for any positive integer greater than 1 by writing n as a perfect power r(d)^k and forming a tree with k branches all equal to b(d). Then a(n) is the number of nodes in b(n).
%e A292127 The first nineteen planted achiral trees are:
%e A292127 o,
%e A292127 (o),
%e A292127 ((o)), (oo),
%e A292127 (((o))), ((oo)),
%e A292127 ((((o)))), (ooo), ((o)(o)), (((oo))),
%e A292127 (((((o))))), ((ooo)), (((o)(o))), ((((oo)))),
%e A292127 ((((((o)))))), (oooo), (((ooo))), ((((o)(o)))), (((((oo))))).
%t A292127 nn=100;
%t A292127 rads=Select[Range[2,nn],GCD@@FactorInteger[#][[All,2]]===1&];
%t A292127 a[1]:=1;a[n_]:=With[{k=GCD@@FactorInteger[n][[All,2]]},1+k*a[Position[rads,n^(1/k)][[1,1]]]];
%t A292127 Array[a,nn]
%Y A292127 Cf. A003238, A007916, A052409, A052410, A061775, A214577, A277576, A277615, A278028, A279614, A279944, A289023.
%K A292127 nonn
%O A292127 1,2
%A A292127 _Gus Wiseman_, Sep 09 2017