A108024 First instance of primes of the form p(p+2)+ k, if they exist, where p and p+2 are prime and k is an even number.

17, 19, 41, 23, 47, 29, 31, 53, 163, 37, 59, 41, 43, 173, 47, 71, 53, 941, 59, 61, 83, 193, 67, 89, 71, 73, 383, 97, 79, 101, 83, 107, 89, 113, 223, 97, 227, 101, 103, 233, 107, 109, 131, 113, 137, 139, 251, 433, 127, 149, 131, 263, 137, 139, 269, 163, 167, 149, 151

Offset: 3

Author

Cino Hilliard, May 31 2005

Keywords

• easy
• nonn

Comments

If p > 3 and k = 6n-2, then p(p+2) + k is composite. This follows from the fact that p and p+2 are both prime iff p = 3m+2 since p = 3m+1 => p+2 = 0 mod 3. Then p(p+2)+6n-2 = 9m^2+18m+8 + 6n-2 = 0 mod 3 composite. Therefore the above seq has no entry for k=10 = 6*2-2 because 8+10 = 0 mod 3. Similarly, if p>3, p=6m+5. As an aside, to test for twin primes > 3 we need only test numbers of the form 6m+5 = 5,11,17,23,29,..

Examples

			3*5+2 = 17,3*5+4=19,5*7+6 = 41.
		

Crossrefs

Cf. A051779.

Programs

PARI

pqpk(n,m,k) = { forstep(k=2,n,2, forprime(x1=3,n, x2=x1+m; p=x1*x2+k; if(isprime(x2)&isprime(p), print1(p",");break; ) ) ) }