A309120 a(n) is the least k > 1 such that n*k is adjacent to a prime.

2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 4, 2, 2, 3, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 6, 3, 6, 5, 2, 2, 2, 2, 4, 2, 2, 2, 4, 5, 4, 2, 2, 2, 4, 2, 2, 3, 2, 2, 2, 2, 6, 2, 2, 3, 2, 2, 2, 3, 4, 3

Offset: 1

Author

Robert Israel, Jul 17 2019

Keywords

• nonn

Comments

If n is odd then a(n) is even.

a(n) exists by Dirichlet's theorem on primes in arithmetic progressions.

Examples

			a(13)=4 because 4*13+1=53 is prime but none of 2*13-1,2*13+1,3*13-1,3*13+1 are primes.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A307833, A104278, A147820.

Programs

Maple

f:= proc(m) local k;
  for k from 2 by 1+(m mod 2) do
    if isprime(k*m-1) or isprime(k*m+1) then return k fi
  od
end proc:
map(f, [$1..100]);

Mathematica

a[n_]:=Module[{k=2},While[Not[PrimeQ[k*n-1]||PrimeQ[k*n+1]],k++];k];
a/@Range[94] (* Ivan N. Ianakiev, Jul 18 2019 *)

PARI

a(n) = my(k=2); while (!isprime(n*k+1) && !isprime(n*k-1), k++); k; \\ Michel Marcus, Jul 19 2019

Formula

a(A104278(n)) > 2 and a(A147820(n)) = 2. - Ivan N. Ianakiev, Jul 18 2019