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A108024
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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.
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0
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 3,1
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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,..
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EXAMPLE
| 3*5+2 = 17,3*5+4=19,5*7+6 = 41.
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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; ) ) ) }
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CROSSREFS
| Cf. A051779.
Sequence in context: A144214 A191043 A129805 * A095081 A144709 A132239
Adjacent sequences: A108021 A108022 A108023 * A108025 A108026 A108027
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KEYWORD
| easy,nonn
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AUTHOR
| Cino Hilliard (hillcino368(AT)gmail.com), May 31 2005
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