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%I #8 Mar 19 2017 19:38:14
%S 5,1,5,5,1,2,4,3,4,0,0,7,4,6,4,4,0,5,5,1,4,1,6,1,9,3,3,7,5,6,5,2,2,8,
%T 2,8,7,4,8,5,7,6,0,4,5,1,8,8,1,1,0,0,2,4,8,3,1,4,3,1,1,0,7,7,6,9,7,3,
%U 5,0,2,9,8,8,6,6,9,4,6,6,3
%N Decimal expansion of a constant relating to the density of Fibonacci integers.
%C Let F(x) be the number of Fibonacci integers, A178772, 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.
%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), pp. 440-457. <a href="http://www.math.dartmouth.edu/~carlp/fibinttalk.pdf">[conference version]</a>
%F 2*zeta(2)*sqrt(zeta(3)/zeta(6)/log(phi)) where phi = (1 + sqrt(5))/2 is the golden ratio.
%e 5.1551243400746440551416193375652282874857604518811002483143110776973502988669...
%t RealDigits[2 Zeta[2] Sqrt[Zeta[3]/Zeta[6]/Log[GoldenRatio]], 10, 81][[1]] (* _Indranil Ghosh_, Mar 19 2017 *)
%o (PARI) phi=(sqrt(5)+1)/2
%o 2*zeta(2)*sqrt(zeta(3)/zeta(6)/log(phi))
%Y Cf. A178772.
%K nonn,cons
%O 1,1
%A _Charles R Greathouse IV_, Aug 31 2016
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