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List of distinct numbers that are powers of odd-indexed Fibonacci numbers or even powers of nonzero even-indexed Fibonacci numbers.
1

%I #12 Jan 05 2025 19:51:42

%S 1,2,4,5,8,9,13,16,25,32,34,64,81,89,125,128,169,233,256,441,512,610,

%T 625,729,1024,1156,1597,2048,2197,3025,3125,4096,4181,6561,7921,8192,

%U 10946,15625,16384,20736,28561,28657,32768,39304,54289,59049,65536,75025,78125

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

%C 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.

%H Amiram Eldar, <a href="/A360772/b360772.txt">Table of n, a(n) for n = 1..1000</a>

%H Hideyuki Ohtsuka, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/FQElemProbFeb2023.pdf">Problem B-1321</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 61, No. 1 (2023), p. 84.

%F Sum_{n>=1} 1/a(n) = 551/126 - 5*sqrt(5)/6.

%t 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]

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

%K nonn

%O 1,2

%A _Amiram Eldar_, Feb 20 2023