

A178772


Fibonacci integers.


6



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
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OFFSET

1,2


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) 440457. (Preprint)


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]] (* JeanFrançois Alcover, Mar 01 2013 *)


CROSSREFS

Cf. A065108, A000045.
Sequence in context: A273885 A174670 A212554 * A296870 A085735 A269393
Adjacent sequences: A178769 A178770 A178771 * A178773 A178774 A178775


KEYWORD

nonn,nice


AUTHOR

T. D. Noe, Jun 11 2010


STATUS

approved



