A363533 Least k such that n*F(k)+1 is prime, where F = A000045 is the Fibonacci sequence, or -1 if no such k exists.

1, 1, 3, 1, 3, 1, 9, 3, 3, 1, 3, 1, 9, 3, 3, 1, 6, 1, 9, 3, 3, 1, 3, 4, 18, 3, 9, 1, 3, 1, 15, 4, 3, 4, 3, 1, 9, 5, 3, 1, 3, 1, 48, 3, 9, 1, 24, 3, 9, 3, 3, 1, 3, 3, 9, 3, 6, 1, 24, 1, 36, 5, 3, 4, 3, 1, 12, 3, 3, 1, 6, 1, 12, 3, 3, 4, 6, 1, 9, 4, 3, 1, 3, 5

Offset: 1

Author

Pontus von Brömssen, Jun 09 2023

Keywords

• nonn

Comments

2 does not appear because F(1) = F(2).

a(n) is divisible by 3 if n >= 3 is odd (unless a(n) = -1), because F(k) is odd (so n*F(k)+1 > 2 is even) when k is not divisible by 3.

Examples

			For n = 17, the least k such that 17*F(k)+1 is prime is k = 6, with 17*F(6)+1 = 17*8+1 = 137, so a(17) = 6.
		

Crossrefs

Cf. A000045, A034693, A124067, A361902, A362376, A363534 (records), A363535 (indices of records), A363536 (first occurrences).

Programs

Mathematica

Array[(k = 1; While[! PrimeQ[# Fibonacci[k] + 1], k++]; k) &, 85] (* Michael De Vlieger, Jun 10 2023 *)

PARI

a(n) = my(k=1); while(!isprime(n*fibonacci(k)+1), k++); k; \\ Michel Marcus, Jun 10 2023

Python

from sympy import isprime, fibonacci
from itertools import count
def A363533(n):
    # Note: the function hangs if a(n) = -1.
    return next(k for k in count(1) if isprime(n*fibonacci(k)+1))

Formula

a(n) = 1 if and only if n+1 is prime.