A275976 Decimal expansion of a constant relating to the density of Fibonacci integers.

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, 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, 5, 0, 2, 9, 8, 8, 6, 6, 9, 4, 6, 6, 3

Offset: 1

Author

Charles R Greathouse IV, Aug 31 2016

Keywords

• nonn
• cons

Comments

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.

Examples

			5.1551243400746440551416193375652282874857604518811002483143110776973502988669...
		

Links

• Florian Luca, Carl Pomerance, Stephan Wagner, Fibonacci Integers, J. Number Theory 131 (2011), pp. 440-457. [conference version]

Crossrefs

Cf. A178772.

Programs

Mathematica

RealDigits[2 Zeta[2] Sqrt[Zeta[3]/Zeta[6]/Log[GoldenRatio]], 10, 81][[1]] (* Indranil Ghosh, Mar 19 2017 *)

PARI

phi=(sqrt(5)+1)/2
2*zeta(2)*sqrt(zeta(3)/zeta(6)/log(phi))

Formula

2*zeta(2)*sqrt(zeta(3)/zeta(6)/log(phi)) where phi = (1 + sqrt(5))/2 is the golden ratio.