A178772 - OEIS

%I #20 Aug 31 2016 14:50:25

%S 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,

%T 27,28,29,30,31,32,33,34,35,36,38,39,40,41,42,44,45,46,47,48,49,50,51,

%U 52,54,55,56,57,58,60,61,62,63,64,65,66,68,69,70,72,75,76,77,78,80,81

%N Fibonacci integers.

%C 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.

%C Integers of the form A065108(n)/A065108(m) for some m and n. - _Charles R Greathouse IV_, Jul 18 2012

%C 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

%H T. D. Noe, <a href="/A178772/b178772.txt">Table of n, a(n) for n = 1..1000</a>

%H Florian Luca, Carl Pomerance, Stephan Wagner, <a href="http://dx.doi.org/10.1016/j.jnt.2010.09.010">Fibonacci Integers</a>, J. Number Theory 131 (2011) 440-457. <a href="http://www.math.dartmouth.edu/~carlp/fibinttalk.pdf">(Preprint)</a>

%t (* 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 *)

%Y Cf. A065108, A000045.

%K nonn,nice

%O 1,2

%A _T. D. Noe_, Jun 11 2010