OFFSET
1,2
COMMENTS
This sequence seems very good at generating primes. Many primes p are generated when a(p)=p. In fact for n<=10000, a(n)=n occurs 1242 times and 1228 of these times n is prime. When n is a composite we often have a(n)<n. However, there are exceptions to this rule and for n<=10000 they are: 323, 377, 442, 1891, 2737, 2834, 3827, 4181, 5777, 6479, 6601, 6721 and 8149 (see A182554).
For n<=1000000 and a(n)=n this sequence generates all primes in the given range (except 5) and produces a prime 99.73% of the time.
It is known that if p is a prime greater than 5 then either F(p-1) or F(p+1) is a multiple of p (see Lawrence and Burton references) and so for those cases we have a(p)=p.
REFERENCES
David M. Burton, Elementary Number Theory, Allyn and Bacon, p. 292, 1980.
LINKS
Dmitry Kamenetsky, Table of n, a(n) for n = 1..10000
D. E. Daykin and L. A. G. Dresel, Factorization of Fibonacci numbers, Fibonacci Quarterly 8:1 (1970), pp. 23-30.
Brian Lawrence, Fibonacci numbers modulo p, Stanford SUMO talk, 2014.
Eric W. Weisstein‘s World of Mathematics, Prime Formulas
Wikipedia, Formula for primes
EXAMPLE
a(2) = max(gcd(2,F(1)),gcd(2,F(3))) = max(1,2)=2.
a(11) = max(gcd(11,F(10)),gcd(11,F(12))) = max(gcd(11,55),gcd(11,144)) = max(11,1) = 11.
a(13) = max(gcd(13,144),gcd(13,377)) = 13.
a(23) = max(gcd(23,17711),gcd(23,46368)) = 23.
MATHEMATICA
Table[Max[GCD[n, Fibonacci[n - 1]], GCD[n, Fibonacci[n + 1]]], {n, 1, 80}] (* Vincenzo Librandi, Jul 20 2015 *)
PROG
(Magma) [Max(Gcd(n, Fibonacci(n-1)), Gcd(n, Fibonacci(n+1))): n in [1..90]]; // Vincenzo Librandi, Jul 20 2015
(PARI) first(m)=vector(m, n, max(gcd(n, fibonacci(n-1)), gcd(n, fibonacci(n+1)))) /* Anders Hellström, Jul 20 2015 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Dmitry Kamenetsky, Jul 20 2015
STATUS
approved