A060385 Largest prime factor of n-th Fibonacci number.

2, 3, 5, 2, 13, 7, 17, 11, 89, 3, 233, 29, 61, 47, 1597, 19, 113, 41, 421, 199, 28657, 23, 3001, 521, 109, 281, 514229, 61, 2417, 2207, 19801, 3571, 141961, 107, 2221, 9349, 135721, 2161, 59369, 421, 433494437, 307, 109441, 28657, 2971215073, 1103

Offset: 3

Author

Labos Elemer, Apr 03 2001

Keywords

• nonn

Comments

For n > 12, Fibonacci(n) is divisible by a primitive prime factor (one not dividing Fibonacci(1), ..., Fibonacci(n-1)). But all primes up to n-2 divide smaller Fibonacci numbers, see A001602, so a(n) >= n-1 for n > 12. This strengthens a more general theorem of Bravo and Luca. - Charles R Greathouse IV, Feb 01 2013

Examples

			F(82) = 2789 * 59369 * 370248451, so a(82) = 370248451.
		

Links

• Tyler Busby, Table of n, a(n) for n = 3..1422 (terms 3..1000 from Charles R Greathouse IV, terms 1001..1408 from Amiram Eldar)
• U. Alfred, On the form of primitive factors of Fibonacci numbers, Fibonacci Quarterly 1:1 (1963), pp. 43-45.
• Jhon J. Bravo and Florian Luca, On the largest prime factor of the k-Fibonacci numbers, arXiv:1210.4101 [math.NT], 2012.
• D. E. Daykin and L. A. G. Dresel, Factorization of Fibonacci numbers, Fibonacci Quarterly 8:1 (1970), pp. 23-30.

Crossrefs

Cf. A000045, A022307, A001605, A060319, A060320, A051694, A193615.

Programs

Magma

[Maximum(PrimeDivisors(Fibonacci(n))): n in [3..50]]; // Vincenzo Librandi, Dec 25 2016

Mathematica

Table[First[Last[FactorInteger[Fibonacci[n]]]], {n, 3, 100}] (* Vladimir Joseph Stephan Orlovsky, Feb 03 2012 *)

PARI

a(n)=my(f=factor(fibonacci(n))[,1]);f[#f] \\ Charles R Greathouse IV, Feb 01 2013

Formula

a(n) >= n - 1 for n > 12, see comments. It is not hard to show that a(n) > 1000 for n > 88. Similarly a(n) > 20641 for n > 120. - Charles R Greathouse IV, Feb 01 2013