

A260222


a(n)=gcd(n,F(n1)), where F(n) is the nth Fibonacci number.


2



1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 11, 1, 1, 1, 1, 2, 1, 1, 19, 1, 3, 2, 1, 1, 1, 1, 1, 2, 29, 1, 31, 1, 3, 2, 1, 1, 1, 1, 1, 2, 41, 1, 1, 1, 3, 2, 1, 1, 7, 1, 1, 2, 1, 1, 1, 1, 3, 2, 59, 1, 61, 1, 1, 2, 1, 1, 1, 1, 3, 2, 71, 1, 1, 1, 1, 2, 1, 13, 79, 1, 3, 2, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,4


COMMENTS

This sequence seems good at generating primes, in particular, twin primes. Many primes p are generated when a(p)=p. In fact for n<=10000, a(n)=n occurs 617 times and 609 of these times n is prime. Furthermore, 275 of these times n is also a twin prime.
For n<=1000000 and a(n)=n this sequence generates 39210 primes (49.95% of primes in the range) and produces a prime 99.75% of the time. At the same time it generates 10864 twin primes, which is 66.50% of all twin primes in the range.
A260228 is a similar sequence that produces more primes.
It is well known that pF(p(p/5)) for every prime p. So a(p) = p for any prime p == 1,4 (mod 5).  ZhiWei Sun, Aug 29, 2015


LINKS

Dmitry Kamenetsky, Table of n, a(n) for n = 1..10000
Z.H. Sun and Z.W. Sun, Fibonacci numbers and Fermat's last theorem, Acta Arithmetica 60(4) (1992), 371388.


EXAMPLE

a(2) = gcd(2,F(1)) = gcd(2,1) = 1.
a(11) = gcd(11,F(10)) = gcd(11,55) = 11.
a(19) = gcd(19,2584) = 19.
a(29) = gcd(29,317811) = 29.


MATHEMATICA

Table[GCD[n, Fibonacci[n1]], {n, 1, 80}] (* Vincenzo Librandi, Jul 20 2015 *)


PROG

(PARI) a(n)=gcd(n, fibonacci(n1))
first(m)=vector(m, n, a(n+1)) /* Anders HellstrÃ¶m, Jul 19 2015 */
(MAGMA) [Gcd(n, Fibonacci(n1)): n in [1..90]]; // Vincenzo Librandi, Jul 20 2015


CROSSREFS

Cf. A104714, A106108, A260228.
Sequence in context: A026536 A046213 A215625 * A181386 A193517 A296554
Adjacent sequences: A260219 A260220 A260221 * A260223 A260224 A260225


KEYWORD

nonn


AUTHOR

Dmitry Kamenetsky, Jul 19 2015


STATUS

approved



