A360772 List of distinct numbers that are powers of odd-indexed Fibonacci numbers or even powers of nonzero even-indexed Fibonacci numbers.

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

Amiram Eldar, Table of n, a(n) for n = 1..1000

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]

Crossrefs

Cf. A000045, A001519, A001906, A103323, A105317, A235383, A254719.

Sequence in context: A065300 A080403 A280050 * A271392 A195364 A279793

Adjacent sequences: A360769 A360770 A360771 * A360773 A360774 A360775

Keyword

nonn

Author

Amiram Eldar, Feb 20 2023

Status

approved