A108024 - OEIS

%I #4 Oct 01 2013 17:58:11

%S 17,19,41,23,47,29,31,53,163,37,59,41,43,173,47,71,53,941,59,61,83,

%T 193,67,89,71,73,383,97,79,101,83,107,89,113,223,97,227,101,103,233,

%U 107,109,131,113,137,139,251,433,127,149,131,263,137,139,269,163,167,149,151

%N 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.

%C 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,..

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

%o (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; ) ) ) }

%Y Cf. A051779.

%K easy,nonn

%O 3,1

%A _Cino Hilliard_, May 31 2005