A361902 Least k such that n+A000045(k) is prime, or -1 if no such k exists.

3, 1, 0, 0, 1, 0, 1, 0, 4, 3, 1, 0, 1, 0, 4, 3, 1, 0, 1, 0, 4, 3, 1, 0, 5, 9, 4, 3, 1, 0, 1, 0, 5, 6, 4, 3, 1, 0, 4, 3, 1, 0, 1, 0, 4, 3, 1, 0, 5, 9, 4, 3, 1, 0, 5, 9, 4, 3, 1, 0, 1, 0, 5, 6, 4, 3, 1, 0, 4, 3, 1, 0, 1, 0, 5, 6, 4, 3, 1, 0, 4, 3, 1, 0, 5, 12, 4

Offset

0,1

Comments

Suggested by A361509.

2 does not appear because A000045(1) = A000045(2).

When n >= 3 and a(n) != -1, a(n) is divisible by 3 if and only if n is odd, because A000045(k) is even if and only if k is divisible by 3.

The least n for which a(n) = -1 is one of 7123, 11009, and 14475. When n is 7123 or 11009, either a(n) > 60000 or a(n) = -1.

Links

Pontus von Brömssen, Table of n, a(n) for n = 0..7122

Robert Gerbicz, Re: Add a Fibonacci number to get a prime, SeqFan mailing list, March 28, 2023.

Formula

a(n) = 0 if and only if n is prime.

a(n) = -1 if n == 14475 (mod m), where m = 2*3*5*7*11*23*31 = 1647030 (see Gerbicz link).

Example

The first Fibonacci number F such that 25+F is prime is F = 34 = A000045(9), so a(25) = 9.

Mathematica

a[n_] := Module[{k = 0}, While[! PrimeQ[n + Fibonacci[k]], k++]; k]; Array[a, 100, 0] (* Amiram Eldar, Mar 30 2023 *)

Prog

(Python)
from sympy import isprime, fibonacci
from itertools import count
def A361902(n):
    # Note: the function hangs if a(n) = -1.
    return next(k for k in count() if isprime(n+fibonacci(k)))
(PARI) a(n) = my(k=0); while (!isprime(n+fibonacci(k)), k++); k; \\ Michel Marcus, Mar 30 2023

Crossrefs

Cf. A000045, A067760, A322004 (negative n), A361509, A361510, A361997 (records), A361998 (indices of records), A361999 (first occurrences).

Sequence in context: A094544 A062734 A336567 * A205531 A269246 A334566

Adjacent sequences: A361899 A361900 A361901 * A361903 A361904 A361905

Keyword

sign

Author

Pontus von Brömssen, Mar 28 2023

Status

approved