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A348348
Smallest k such that the numbers j*k - 1 and j*k + 1 are prime for 1 <= j <= n.
5
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).
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
KEYWORD
nonn,more
AUTHOR
EXTENSIONS
a(5), a(6) from Jon E. Schoenfield, Nov 14 2021
a(7) from Klaus Muuss, Jul 01 2022
STATUS
approved