

A305534


Index of the smallest prime in the nFibonacci sequence, or the Lucas U(n,1) sequence.


0



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, 5, 5, 5479, 71, 3, 2, 17, 11, 3, 2, 37, 2, 31, 5, 11, 2, 5
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

Smallest k such that the kth 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, ...)
All terms are primes, since if a divides b, then the ath term of the nFibonacci sequence also divides the bth term of the nFibonacci sequence.
Corresponding primes are 2, 2, 3, 17, 5, 37, 7, 4289, 726120289954448054047428229, 101, 11, 21169, 13, 197, 82088569942721142820383601, 257, 17, 34539049, 19, 401, ...
a(n) = 2 if and only if n is prime.
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).
a(34) > 1024, does a(n) exist for all n >= 1? (However, 17 is the only prime in the first 1024 terms of the 4Fibonacci sequence, and it seems that 17 is the only prime in the 4Fibonacci sequence.)
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.
a(49) > 20000, if it exists.  Giovanni Resta, Jun 06 2018


LINKS

Table of n, a(n) for n=1..48.


PROG

(PARI) b(n, k)=([n, 1; 1, 0]^k)[1, 2]
a(n)=for(k=1, 2^12, if(ispseudoprime(b(n, k)), return(k)))


CROSSREFS

Cf. A001605, A096650, A209493, which are the indices of the primes in the nFibonacci sequence for n = 1, 2, 3.
Cf. A005478, A086383, A201001, which are the primes in the nFibonacci sequence for n = 1, 2, 3.
Cf. A000045, A000129, A006190, A001076, A052918, A005668, A054413, A041025, A099371, A041041, A049666, A041061 (the nFibonacci sequence for n = 1 to 12).
Cf. A302990 (for nstep Fibonacci sequence instead of nFibonacci sequence).
Sequence in context: A340300 A245070 A270226 * A248138 A049234 A299351
Adjacent sequences: A305531 A305532 A305533 * A305535 A305536 A305537


KEYWORD

nonn,more


AUTHOR

Eric Chen, Jun 04 2018


EXTENSIONS

a(34)a(48) from Giovanni Resta, Jun 06 2018


STATUS

approved



