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%I #38 Jun 06 2018 08:40:14
%S 3,2,2,3,2,3,2,5,29,3,2,5,2,3,23,3,2,7,2,3,29,19,2,3,83,3,53,19,2,5,2,
%T 5,5,5479,71,3,2,17,11,3,2,37,2,31,5,11,2,5
%N Index of the smallest prime in the n-Fibonacci sequence, or the Lucas U(n,-1) sequence.
%C Smallest k such that the k-th Fibonacci polynomial evaluated at x=n is prime. (The first few Fibonacci polynomials are 1, x, x^2 + 1, x^3 + 2*x, x^4 + 3*x^2 + 1, x^5 + 4*x^3 + 3*x, ...)
%C All terms are primes, since if a divides b, then the a-th term of the n-Fibonacci sequence also divides the b-th term of the n-Fibonacci sequence.
%C Corresponding primes are 2, 2, 3, 17, 5, 37, 7, 4289, 726120289954448054047428229, 101, 11, 21169, 13, 197, 82088569942721142820383601, 257, 17, 34539049, 19, 401, ...
%C a(n) = 2 if and only if n is prime.
%C a(n) = 3 if and only if n^2 + 1 is prime (A005574), except n=2 (since 2 is the only prime p such that p^2 + 1 is also prime).
%C a(34) > 1024, does a(n) exist for all n >= 1? (However, 17 is the only prime in the first 1024 terms of the 4-Fibonacci sequence, and it seems that 17 is the only prime in the 4-Fibonacci sequence.)
%C a(35)..a(48) = 71, 3, 2, 17, 11, 3, 2, 37, 2, 31, 5, 11, 2, 5, a(50)..a(54) = 11, 11, 23, 2, 3, a(56) = 3, a(58)..a(75) = 5, 2, 47, 2, 5, 311, 13, 233, 3, 2, 5, 11, 5, 2, 7, 2, 3, 5. Unknown terms a(34), a(49), a(55), a(57), exceed 1024, if they exist.
%C a(49) > 20000, if it exists. - _Giovanni Resta_, Jun 06 2018
%o (PARI) b(n,k)=([n,1;1,0]^k)[1,2]
%o a(n)=for(k=1,2^12,if(ispseudoprime(b(n,k)),return(k)))
%Y Cf. A001605, A096650, A209493, which are the indices of the primes in the n-Fibonacci sequence for n = 1, 2, 3.
%Y Cf. A005478, A086383, A201001, which are the primes in the n-Fibonacci sequence for n = 1, 2, 3.
%Y Cf. A000045, A000129, A006190, A001076, A052918, A005668, A054413, A041025, A099371, A041041, A049666, A041061 (the n-Fibonacci sequence for n = 1 to 12).
%Y Cf. A302990 (for n-step Fibonacci sequence instead of n-Fibonacci sequence).
%K nonn,more
%O 1,1
%A _Eric Chen_, Jun 04 2018
%E a(34)-a(48) from _Giovanni Resta_, Jun 06 2018
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