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%I #23 Jan 15 2015 12:07:28
%S 5,13,21,34,55,144,610,2584,6765,10946,46368,196418,832040,14930352,
%T 267914296,1134903170,4807526976,365435296162,1548008755920,
%U 117669030460994,498454011879264,2111485077978050,160500643816367088,12200160415121876738,51680708854858323072
%N Fibonacci numbers k for which the difference between k and the largest prime less than k is also prime.
%C a(n) - p = q, where a(n) is a Fibonacci number, p is the largest prime less than a(n), and q is also prime.
%C The only terms that are primes are 5 and 13, since there are no other Fibonacci numbers that are twin primes: see the MacKinnon and Gagola link. - _Robert Israel_, Jan 13 2015
%H N. MacKinnon and S. M. Gagola, Jr., <a href="http://www.jstor.org/stable/2695779">Fibonacci twin primes (solution to problem 10844)</a>, American Mathematical Monthly 109, No. 1 (Jan., 2002), 78.
%e For n = 1: a(1) = 5, 5 - 3 = 2.
%e For n = 4: a(4) = 34, 34 - 31 = 3.
%e For n = 7: a(7) = 610, 610 - 607 = 3.
%e For n = 11: a(11) = 46368, 46368 - 46351 = 17.
%p select(t -> isprime(t - prevprime(t)), [seq(combinat:-fibonacci(n),n=4..1000)]); # _Robert Israel_, Dec 16 2014
%t Select[ Fibonacci@ Range[4, 100], PrimeQ[# - NextPrime[#, -1]] &]
%o (PARI) for(n=1,100,f=fibonacci(n);if(f>2&&isprime(f-precprime(f-1)),print1(f,", "))) \\ _Derek Orr_, Dec 30 2014
%Y Cf. A180422, A000045.
%K nonn,easy
%O 1,1
%A _Carlos Eduardo Olivieri_, Dec 16 2014