A382246 Smallest number k such that k^n - 6 is prime.

8, 3, 2, 5, 5, 5, 19, 85, 7, 5, 19, 275, 23, 43, 53, 455, 65, 23, 23, 175, 7, 65, 47, 295, 7, 143, 49, 115, 23, 355, 185, 305, 7, 55, 319, 85, 113, 25, 329, 505, 25, 187, 205, 25, 295, 437, 17, 2285, 7, 583, 35, 1375, 5, 7, 35, 895, 235, 277, 197, 695, 203, 145, 43, 35, 437, 215

Offset

1,1

Comments

No term k in the sequence can be divisible by 2 or 3. Except for the special case a(1)-a(3), where the result of k^n - 6 is either the prime number 2 or 3.

If n is a multiple of 4, the only valid terms of k are those ending in a 5.

Empirical analysis suggests that the terms are typically prime or semiprime.

Links

Jakub Buczak, Table of n, a(n) for n = 1..500

Example

a(1) = 8, because 8^1 - 6 = 2, which is prime.
a(4) = 5, because 5^4 - 6 = 619, which is prime.

Prog

(Python)
from sympy import isprime
def a(n):
    k = 1
    while (n>1 and k not in [2, 3] and (k%2==0 or k%3==0)) or not isprime(k**n-6):
        k += 1
    return k
(PARI) a(n) = my(k=1); while (!isprime(k^n-6), k++); k; \\ Michel Marcus, Mar 19 2025

Crossrefs

Cf. A380905

Cf. A028879 (a(2)), A239414 (a(6)) for the first term.

Sequence in context: A119277 A273556 A346713 * A154014 A063568 A302138

Adjacent sequences: A382243 A382244 A382245 * A382247 A382248 A382249

Keyword

nonn

Author

Jakub Buczak, Mar 19 2025

Status

approved