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A260222
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a(n)=gcd(n,F(n-1)), where F(n) is the n-th Fibonacci number.
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2
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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
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OFFSET
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1,4
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COMMENTS
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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
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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PROG
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(PARI) a(n)=gcd(n, fibonacci(n-1))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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