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A260222 a(n)=gcd(n,F(n-1)), where F(n) is the n-th 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 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

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), 371-388.

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[n-1]], {n, 1, 80}] (* Vincenzo Librandi, Jul 20 2015 *)

PROG

(PARI) a(n)=gcd(n, fibonacci(n-1))

first(m)=vector(m, n, a(n+1)) /* Anders Hellström, Jul 19 2015 */

(MAGMA) [Gcd(n, Fibonacci(n-1)): 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

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Last modified April 5 04:23 EDT 2020. Contains 333238 sequences. (Running on oeis4.)