A302990 a(n) = index of first odd prime number in the (n-th)-order Fibonacci sequence Fn, or 0 if no such index exists.

0, 0, 4, 6, 9, 10, 40, 14, 17, 19, 361, 23, 90, 26, 373, 47, 288, 34, 75, 38, 251, 43, 67, 47, 74, 310, 511, 151534, 57, 20608, 1146, 62, 197, 94246, 9974, 287, 271172, 758

Offset: 0

Author

Jacques Tramu, Apr 17 2018

Keywords

• nonn
• more

Comments

Fn is defined by: Fn(0) = Fn(1) = ... = Fn(n-2) = 0, Fn(n-1) = 1, and Fn(k+1) = Fn(k) + Fn(k-1) + ... + Fn(k-n+1).

In general, Fn(k) is odd iff k == -1 or -2 (mod n+1), therefore a(n) = k*(n+1) - (1 or 2) for all n. Since Fn(n-1) = F(n) = 1, we must have a(n) >= 2n. Since Fn(k) = 2^(k-n) for n <= k < 2n, Fn(2n) = 2^n-1, so a(n) = 2n exactly for the Mersenne prime exponents A000043, while a(n) = 2n+1 when n is not in A000043 but n+1 is in A050414. - M. F. Hasler, Apr 18 2018

Further terms of the sequence: a(38) > 62000, a(39) > 72000, a(40) = 285, a(41) > 178000, a(42) = 558, a(44) = 19529, a(46) = 33369, a(47) = 239, a(48) = 6368, a(53) = 2860, a(54) = 2418, a(58) = 176, a(59) = 18418, a(60) = 1463, a(61) = 122, a(62) = 8755, a(63) = 5118, a(64) = 25089, a(65) = 988, a(66) = 333, a(67) = 406, a(70) = 1632, a(74) = 374, a(76) = 13704, a(77) = 4991, a(86) = 347, a(89) = 178, a(92) = 1114, a(93) = 187, a(98) = 395, a(100) > 80000; a(n) > 10^4 for all other n up to 100. - Jacques Tramu and M. F. Hasler, Apr 18 2018

Examples

			a(2) = 4 because F2 (Fibonacci) = 0, 1, 1, 2, 3, 5, 8, ... and F2(4) = 3 is prime.
a(3) = 6 because F3 (tribonacci) = 0, 0, 1, 1, 2, 4, 7, 13, ... and F3(6) = 7 is prime.
a(4) = 9 because F4 (tetranacci) = 0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, ...  and F4(9) = 29 is prime.
From _M. F. Hasler_, Apr 18 2018: (Start)
We see that Fn(k) = 2^(k-n) for n <= k < 2n and thus Fn(2n) = 2^n-1, so a(n) = 2n exactly for the Mersenne prime exponents A000043.
a(n) = 2n + 1 when 2^(n+1) - 3 is prime (n+1 in A050414) but 2^n-1 is not, i.e., n = 4, 8, 9, 11, 21, 23, 28, 93, 115, 121, 149, 173, 212, 220, 232, 265, 335, 451, 544, 688, 693, 849, 1735, ...
For other primes we have: a(29) = 687*30 - 2, a(37) = 20*38 - 2, a(41) > 10^4, a(43) > 10^4, a(47) = 5*48 - 1, a(53) = 53*54 - 2, a(59) = 307*60 - 2, a(67) = 6*67 - 1. (End)
		

Crossrefs

Cf. A000045 (F2), A000073 (F3), A000078 (F4), A001591 (F5), A001592 (F6), A122189(F7), A079262 (F8), A104144 (F9), A122265 (F10).

(According to the definition, F0 = A000004 and F1 = A000012.)

Cf. A001605 (indices of prime numbers in F2).

Cf. A000043, A050414.

Programs

PARI

A302990(n,L=oo,a=vector(n+1,i,if(i1 && for(i=-2+2*n+=1,L, ispseudoprime(a[i%n+1]=2*a[(i-1)%n+1]-a[i%n+1]) && return(i))} \\ Testing primality only for i%n>n-3 is not faster, even for large n. - M. F. Hasler, Apr 17 2018; improved Apr 18 2018

Formula

a(n) == -1 or -2 (mod n+1). a(n) >= 2n, with equality iff n is in A000043. a(n) <= 2n+1 for n+1 in A050414. - M. F. Hasler, Apr 18 2018

Extensions

a(29) from Jacques Tramu, Apr 19 2018

a(33) from Daniel Suteu, Apr 20 2018

a(36) from Jacques Tramu, Apr 25 2018