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A260228
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a(n) = max(gcd(n,F(n-1)),gcd(n,F(n+1))), where F(n) is the n-th Fibonacci number.
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2
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1, 2, 3, 2, 1, 1, 7, 2, 3, 2, 11, 1, 13, 2, 3, 2, 17, 1, 19, 2, 3, 2, 23, 1, 1, 2, 3, 2, 29, 1, 31, 2, 3, 2, 1, 1, 37, 2, 3, 2, 41, 1, 43, 2, 3, 2, 47, 1, 7, 2, 3, 2, 53, 1, 1, 2, 3, 2, 59, 1, 61, 2, 21, 2, 1, 1, 67, 2, 3, 2, 71, 1, 73, 2, 3, 2, 1, 13, 79, 2, 3, 2, 83, 1, 1, 2, 3, 2, 89, 1, 1, 2, 3, 2, 1, 1, 97, 2, 33, 2
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OFFSET
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1,2
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COMMENTS
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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.
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REFERENCES
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David M. Burton, Elementary Number Theory, Allyn and Bacon, p. 292, 1980.
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LINKS
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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.
Ron Knott, Factorisation of the first 300 Fibonacci numbers
Brian Lawrence, Fibonacci numbers modulo p, Stanford SUMO talk, 2014.
Eric W. Weisstein, MathWorld: Prime Formulas
Wikipedia, Formula for primes
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EXAMPLE
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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.
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MATHEMATICA
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Table[Max[GCD[n, Fibonacci[n - 1]], GCD[n, Fibonacci[n + 1]]], {n, 1, 80}] (* Vincenzo Librandi, Jul 20 2015 *)
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PROG
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(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 */
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CROSSREFS
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Cf. A106108, A182554, A260222.
Sequence in context: A082868 A219539 A154556 * A316657 A277914 A126626
Adjacent sequences: A260225 A260226 A260227 * A260229 A260230 A260231
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KEYWORD
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nonn
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AUTHOR
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Dmitry Kamenetsky, Jul 20 2015
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STATUS
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approved
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