A348348 Smallest k such that the numbers j*k - 1 and j*k + 1 are prime for 1 <= j <= n.

4, 6, 6, 21968100, 100803789240, 683751016938990, 1651735848676253340

Offset

1,1

Comments

The following heuristic argument suggests that a(n) exists for all n: For large (random) k and a specific j <= n, the probability that both j*k - 1 and j*k + 1 are prime should be of the order 1/(log k)^2 (a slight twist of the first Hardy-Littlewood conjecture). Assuming independence between different j, the probability that this holds for 1 <= j <= n is of the order 1/(log k)^(2*n). Since the sum over k of 1/(log k)^(2*n) diverges, this should hold for infinitely many k by the second Borel-Cantelli lemma (assuming independence between different k).

Links

Table of n, a(n) for n=1..7.

Wikipedia, Borel-Cantelli lemma

Wikipedia, Twin prime

Example

a(1) = A014574(1) = 4.
a(2) = A066388(1) = 6.
a(3) = A118859(1) = 6.
a(4) = A118860(1) = 21968100.
a(5) = A349321(1) = 100803789240.

Prog

(Python)
from sympy import isprime, nextprime
def A348348(n):
    p = 2
    while 1:
        p_next = nextprime(p)
        if p_next == p+2 and all(isprime(j*(p+1)-1) and isprime(j*(p+1)+1) for j in range(2, n+1)):
            return p+1
        p = p_next
(PARI) isok(k, n) = for (j=1, n, if (!isprime(j*k-1) || !isprime(j*k+1), return(0))); return(1);
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Jul 01 2022

Crossrefs

Cf. A014574, A066388, A118859, A118860, A349321.

Sequence in context: A365084 A082237 A348958 * A042976 A090142 A086237

Adjacent sequences: A348345 A348346 A348347 * A348349 A348350 A348351

Keyword

nonn,more

Author

Pontus von Brömssen, Oct 13 2021

Extensions

a(5), a(6) from Jon E. Schoenfield, Nov 14 2021

a(7) from Klaus Muuss, Jul 01 2022

Status

approved