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

3,1

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,..

Links

Table of n, a(n) for n=3..61.

Example

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

Prog

(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; ) ) ) }

Crossrefs

Cf. A051779.

Sequence in context: A129805 A289492 A262286 * A095081 A243437 A144709

Adjacent sequences: A108021 A108022 A108023 * A108025 A108026 A108027

Keyword

easy,nonn

Author

Cino Hilliard, May 31 2005

Status

approved