A357746 Primes p such that the least k for which k*p + 1 is prime is also the least k for which k*p - 1 is prime.

47, 103, 107, 283, 313, 347, 397, 773, 787, 907, 1051, 1117, 1319, 1433, 1823, 2027, 2153, 2203, 2287, 2333, 2347, 2381, 2909, 3221, 3257, 3673, 3923, 3929, 4129, 4153, 4217, 4547, 4597, 4657, 4721, 4969, 5023, 5387, 5407, 5693, 5717, 5827, 5881, 6373, 6781, 6863, 6997

Offset: 1

Author

Karl-Heinz Hofmann, Jan 01 2023

Keywords

• nonn

Comments

If A035096(n) = A216568(n) the n-th prime is a term. Here k*p must be the composite number sandwiched between a pair of twin primes, so by Wilson's theorem, k must be a multiple of 6.

Examples

			a(1) = 47: 47*6 + 1 = 283 (a prime), 47*6 - 1 = 281 (also a prime), and no k < 6 gives a prime as the result for both formulas.
		

Links

• Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000
• Wikipedia, Wilson's theorem.

Crossrefs

Cf. A000040, A014574, A001359, A006512, A035096, A216568.

Programs

Mathematica

q[p_] := Module[{k = 1, r}, While[! Or @@ (r = PrimeQ[k*p + {-1, 1}]), k++]; And @@ r]; Select[Prime[Range[900]], q] (* Amiram Eldar, Jan 01 2023 *)

PARI

isk(p, x) = my(k=1); while (!isprime(k*p+x), k++); k;
isok(p) = if (isprime(p), isk(p, +1) == isk(p, -1)); \\ Michel Marcus, Jan 01 2023

Python

from sympy import sieve, isprime
def leastk(p, plusminus):
    k=1
    while not isprime(k * p + plusminus): k += 1
    return k
print([p for p in sieve[1:1000] if leastk(p, 1) == leastk(p, -1)])