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%I #41 Sep 08 2022 08:46:13
%S 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,
%T 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,
%U 1,1,1,3,2,71,1,1,1,1,2,1,13,79,1,3,2,1
%N a(n)=gcd(n,F(n-1)), where F(n) is the n-th Fibonacci number.
%C 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.
%C 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.
%C A260228 is a similar sequence that produces more primes.
%C It is well known that p|F(p-(p/5)) for every prime p. So a(p) = p for any prime p == 1,4 (mod 5). - _Zhi-Wei Sun_, Aug 29, 2015
%H Dmitry Kamenetsky, <a href="/A260222/b260222.txt">Table of n, a(n) for n = 1..10000</a>
%H Z.-H. Sun and Z.-W. Sun, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa60/aa6046.pdf">Fibonacci numbers and Fermat's last theorem</a>, Acta Arithmetica 60(4) (1992), 371-388.
%e a(2) = gcd(2,F(1)) = gcd(2,1) = 1.
%e a(11) = gcd(11,F(10)) = gcd(11,55) = 11.
%e a(19) = gcd(19,2584) = 19.
%e a(29) = gcd(29,317811) = 29.
%t Table[GCD[n, Fibonacci[n-1]], {n, 1, 80}] (* _Vincenzo Librandi_, Jul 20 2015 *)
%o (PARI) a(n)=gcd(n,fibonacci(n-1))
%o first(m)=vector(m,n,a(n+1)) /* _Anders Hellström_, Jul 19 2015 */
%o (Magma) [Gcd(n,Fibonacci(n-1)): n in [1..90]]; // _Vincenzo Librandi_, Jul 20 2015
%Y Cf. A104714, A106108, A260228.
%K nonn
%O 1,4
%A _Dmitry Kamenetsky_, Jul 19 2015
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