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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 (list; graph; refs; listen; history; text; internal format)
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).

LINKS

Table of n, a(n) for n=1..77.

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]

CROSSREFS

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

Sequence in context: A061339 A073933 A056792 * A227861 A294991 A300118

Adjacent sequences:  A292124 A292125 A292126 * A292128 A292129 A292130

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 09 2017

STATUS

approved

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Last modified July 23 22:47 EDT 2019. Contains 325278 sequences. (Running on oeis4.)