A381532 Smallest integer k>0 such that prime(n) + k*prime(n+1) is prime.

1, 2, 2, 2, 2, 2, 6, 6, 4, 8, 4, 6, 2, 2, 10, 10, 2, 6, 12, 6, 4, 6, 4, 2, 14, 2, 2, 6, 6, 2, 2, 6, 6, 6, 20, 6, 4, 8, 4, 16, 2, 2, 2, 2, 8, 10, 4, 2, 6, 6, 6, 14, 2, 4, 10, 6, 2, 6, 2, 6, 18, 2, 2, 2, 2, 12, 10, 2, 6, 6, 4, 2, 22, 4, 6, 10, 12, 6, 8, 8, 12, 2

Offset: 1

Author

Jean-Marc Rebert, Mar 07 2025

Keywords

• nonn

Examples

			a(1)= 1, because 2 and 3 are consecutive primes and 2 + 1*3 = 5 is prime, and no lesser number has this property.
 p + k*q, where p and q are consecutive primes
 2 + 1* 3 =   5 is prime;
 3 + 2* 5 =  13 is prime;
 5 + 2* 7 =  19 is prime;
 7 + 2*11 =  29 is prime;
		

Crossrefs

Cf. A129919 (resulting primes), A175914 (primes for which k=2), A368691 (primes for which k=4).

Programs

Mathematica

Do[k=0;Until[PrimeQ[Prime[n]+k*Prime[n+1]],k++];a[n]=k,{n,82}];Array[a,82] (* James C. McMahon, Mar 28 2025 *)

PARI

a(n) = my(p=prime(n), q=nextprime(p+1), k=1); while (!isprime(p+k*q), k++); k; \\ Michel Marcus, Mar 09 2025