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URL: https://oeis.org/A360772

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A360772
List of distinct numbers that are powers of odd-indexed Fibonacci numbers or even powers of nonzero even-indexed Fibonacci numbers.
1
1, 2, 4, 5, 8, 9, 13, 16, 25, 32, 34, 64, 81, 89, 125, 128, 169, 233, 256, 441, 512, 610, 625, 729, 1024, 1156, 1597, 2048, 2197, 3025, 3125, 4096, 4181, 6561, 7921, 8192, 10946, 15625, 16384, 20736, 28561, 28657, 32768, 39304, 54289, 59049, 65536, 75025, 78125
OFFSET
1,2
COMMENTS
Ohtsuka's (2023) problem does not include 1, and includes the even powers of 8 twice (once as powers of Fibonacci(6) = 8 and once as powers of Fibonacci(3) = 2). The sum of reciprocals in this case is (61 - 15*sqrt(5))/18.
LINKS
Hideyuki Ohtsuka, Problem B-1321, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 61, No. 1 (2023), p. 84.
FORMULA
Sum_{n>=1} 1/a(n) = 551/126 - 5*sqrt(5)/6.
MATHEMATICA
seq[max_] := Module[{s = {1}, k = 3, f, d}, While[(f = Fibonacci[k]) <= max, If[k != 6, d = 2 - Mod[k, 2]; s = Join[s, f^Range[d, Floor[Log[f, max]], d]]]; k++]; Sort[s]]; seq[10^5]
KEYWORD
nonn
AUTHOR
Amiram Eldar, Feb 20 2023
STATUS
approved