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A060385
Largest prime factor of n-th Fibonacci number.
10
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
3,1
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
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.
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
EXAMPLE
F(82) = 2789 * 59369 * 370248451, so a(82) = 370248451.
MATHEMATICA
Table[First[Last[FactorInteger[Fibonacci[n]]]], {n, 3, 100}] (* Vladimir Joseph Stephan Orlovsky, Feb 03 2012 *)
PROG
(PARI) a(n)=my(f=factor(fibonacci(n))[, 1]); f[#f] \\ Charles R Greathouse IV, Feb 01 2013
(Magma) [Maximum(PrimeDivisors(Fibonacci(n))): n in [3..50]]; // Vincenzo Librandi, Dec 25 2016
KEYWORD
nonn
AUTHOR
Labos Elemer, Apr 03 2001
STATUS
approved