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

1,1

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.

Links

Table of n, a(n) for n=1..81.

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

Formula

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

Example

5.1551243400746440551416193375652282874857604518811002483143110776973502988669...

Mathematica

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

Prog

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

Crossrefs

Cf. A178772.

Sequence in context: A073226 A309282 A021198 * A306577 A143969 A198366

Adjacent sequences: A275973 A275974 A275975 * A275977 A275978 A275979

Keyword

nonn,cons

Author

Charles R Greathouse IV, Aug 31 2016

Status

approved