A178772 Fibonacci integers.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 72, 75, 76, 77, 78, 80, 81

Offset: 1

Author

T. D. Noe, Jun 11 2010

Keywords

• nonn
• nice

Comments

A Fibonacci integer is a number that can be written as the product and/or quotient of Fibonacci numbers (A000045). For example, 33 is a Fibonacci integer because Fib(10) * Fib(4) / Fib(5) = 33. Of the numbers up to 100, only 8 are not Fibonacci integers: 37, 43, 53, 59, 67, 71, 73, 74, 79, 83, 86, and 97. See A178762 for the prime numbers in this sequence.

Integers of the form A065108(n)/A065108(m) for some m and n. - Charles R Greathouse IV, Jul 18 2012

Let F(x) be the number of terms of this sequence less than or equal to x. Then exp(c*sqrt(log x) - (log x)^e) < F(x) < exp(c*sqrt(log x) + (log x)^(1/6 + e)) for any e > 0, where c is this constant. Luca, Pomerance, & Wagner conjecture that 1/6 can be replaced by 0, and note that it can be replaced by 1/8 on a strong form of the abc conjecture. - Charles R Greathouse IV, Aug 31 2016

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Florian Luca, Carl Pomerance, Stephan Wagner, Fibonacci Integers, J. Number Theory 131 (2011) 440-457. (Preprint)

Crossrefs

Cf. A065108, A000045.

Programs

Mathematica

(* This naive program being quite slow, some terms are precomputed. *) max = Fibonacci[m = 20]; Clear[f]; Do[ f[n] = True, {n, {23, 31, 46, 62, 69, 92, 93}}]; f[] = False; Do[ f[fn = Fibonacci[n]] = True; f[fk = Fibonacci[k]] = True; If[ fn*fk < max, f[fn*fk] = True]; If[ IntegerQ[fk/fn] && fk/fk < max, f[fk/fn] = True], {n, 2, m}, {k, n, m}]; fp[] := (cnt = 0; Do[ If [f[n] && f[k], If[ n*k < max, f[n*k] = True; cnt++]; If[ IntegerQ[k/n], f[k/n] = True; cnt++]], {n, 1, max}, {k, n+1, max}]; Print[cnt, " Fibonacci integers"]; cnt); FixedPoint[fp, 0]; Reap[ Do[ If[ f[n], Sow[n]], {n, 1, 100}]][[2, 1]] (* Jean-François Alcover, Mar 01 2013 *)