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A030426
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a(n) = Fibonacci(prime(n)).
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22
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1, 2, 5, 13, 89, 233, 1597, 4181, 28657, 514229, 1346269, 24157817, 165580141, 433494437, 2971215073, 53316291173, 956722026041, 2504730781961, 44945570212853, 308061521170129, 806515533049393, 14472334024676221, 99194853094755497, 1779979416004714189
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OFFSET
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1,2
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COMMENTS
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Except for Fibonacci(4) = 3, if Fibonacci(n) is prime, then n is also prime. However, if n is prime, Fibonacci(n) might be composite, as, for example, Fibonacci(19) = 4181 = 37 * 113. - Alonso del Arte, Jan 28 2014
The values are pairwise relatively prime because gcd(Fib(m), Fib(n)) = Fib(gcd(m, n)) and this equals Fib(1) = 1 when m!=n are prime numbers. - Lee A. Newberg, May 05 2023
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LINKS
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Michel Bataille, Problem 90.G, Problem Corner, The Mathematical Gazette, Vol. 90, No. 518 (2006), p. 354; Solution, ibid., Vol. 91, No. 520 (2007), pp. 160-161.
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FORMULA
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a(n) == 1 (mod prime(n)) if prime(n) == 1, 4 (mod 5).
a(n) == -1 (mod prime(n)) if prime(n) == 2, 3 (mod 5). (End)
a(n) == Sum_{k=0..floor((prime(n)-1)/2)} (-1)^k * binomial(2*k,k) (mod prime(n)) (Bataille, 2006). - Amiram Eldar, Jul 02 2023
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MAPLE
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with(combinat); for i from 1 to 50 do fibonacci(ithprime(i)); od;
# second Maple program:
a:= n-> (<<0|1>, <1|1>>^ithprime(n))[1, 2]:
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MATHEMATICA
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PROG
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(GAP) a:=List(Filtered([1..100], IsPrime), i->Fibonacci(i));; Print(a); # Muniru A Asiru, Dec 29 2018
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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John C. Hallyburton, Jr. (jhallyburton(AT)mx1.AspenRes.Com)
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STATUS
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approved
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