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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).
0
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, 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, 11, 13, 7, 11, 10, 11, 12, 13, 11, 13, 10, 12, 14, 8, 12, 11
OFFSET
1,2
COMMENTS
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).
EXAMPLE
The first nineteen planted achiral trees are:
o,
(o),
((o)), (oo),
(((o))), ((oo)),
((((o)))), (ooo), ((o)(o)), (((oo))),
(((((o))))), ((ooo)), (((o)(o))), ((((oo)))),
((((((o)))))), (oooo), (((ooo))), ((((o)(o)))), (((((oo))))).
MATHEMATICA
nn=100;
rads=Select[Range[2, nn], GCD@@FactorInteger[#][[All, 2]]===1&];
a[1]:=1; a[n_]:=With[{k=GCD@@FactorInteger[n][[All, 2]]}, 1+k*a[Position[rads, n^(1/k)][[1, 1]]]];
Array[a, nn]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 09 2017
STATUS
approved