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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).
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%I #10 May 04 2018 22:42:29

%S 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 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 11,13,7,11,10,11,12,13,11,13,10,12,14,8,12,11

%N 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 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 The first nineteen planted achiral trees are:

%e o,

%e (o),

%e ((o)), (oo),

%e (((o))), ((oo)),

%e ((((o)))), (ooo), ((o)(o)), (((oo))),

%e (((((o))))), ((ooo)), (((o)(o))), ((((oo)))),

%e ((((((o)))))), (oooo), (((ooo))), ((((o)(o)))), (((((oo))))).

%t nn=100;

%t rads=Select[Range[2,nn],GCD@@FactorInteger[#][[All,2]]===1&];

%t a[1]:=1;a[n_]:=With[{k=GCD@@FactorInteger[n][[All,2]]},1+k*a[Position[rads,n^(1/k)][[1,1]]]];

%t Array[a,nn]

%Y Cf. A003238, A007916, A052409, A052410, A061775, A214577, A277576, A277615, A278028, A279614, A279944, A289023.

%K nonn

%O 1,2

%A _Gus Wiseman_, Sep 09 2017