A139602 The smallest prime p that makes the pair p+/-6n both primes while no other pair of p+/-6k+6*n, 0<k<n both primes.

11, 19, 61, 43, 97, 163, 191, 229, 283, 223, 743, 991, 541, 457, 877, 1327, 1049, 1321, 1733, 1307, 1987, 6011, 2971, 5153, 2029, 8693, 2551, 4789, 5407, 2129, 6473, 4481, 4889, 4217, 7951, 11743, 13789, 9851, 7253, 11491, 20393, 17231, 9749, 20747, 6599, 13873, 16369, 15461, 17123, 13451, 9967, 26959, 21089, 41863, 27437, 26003, 40189, 18661, 16673, 64693

Offset

1,1

Comments

Dickson's conjecture implies that this sequence is infinite. - Charles R Greathouse IV, Mar 22 2011

2a(n) = (p-6n) + (p+6n).

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000

Example

For n = 1, 11-6n=5, 11+6n=17, are both primes, and for any prime number p smaller than 11, it is impossible that p-6 is prime;
For n = 2, 19-6n=7, 19+6n=31, are both primes,while 19+6*1=25 is not prime. For primes p<19, either p+/-6 are prime pairs, or p+/-12 are not a prime pair;
...
for n = 6, 163-6n=127, 163+6n=199, are both primes,while 163+6*k, k=1,2,4 and 163-6*k, k=3,5 are not primes. For primes p<163, either exists prime pair p+/-6k, 0<k<6, or p+/-36 are not a prime pair.

Mathematica

BothPrime[p_, k_] := PrimeQ[p + 6*k] && PrimeQ[p - 6*k]; f[n_] := Module[{p, k}, p = Prime[PrimePi[6*n] + 1]; While[k = 1; While[k < n && ! BothPrime[p, k], k++]; k < n || ! BothPrime[p, n],  p = NextPrime[p]]; p]; Table[f[n], {n, 60}]

Prog

(PARI) a(n)=forprime(p=2, default(primelimit), if(!isprime(p-6*n) || !isprime(p+6*n), next); for(k=1, n-1, if(isprime(p-6*k) && isprime(p+6*k), next(2))); return(p)) \\ Charles R Greathouse IV, Mar 22 2011

Crossrefs

Sequence in context: A138355 A265802 A178385 * A080789 A057770 A080788

Adjacent sequences: A139599 A139600 A139601 * A139603 A139604 A139605

Keyword

nonn

Author

Lei Zhou, Mar 22 2011

Status

approved