A275021 Pairs of primes p, p+4 such that p-2 and p+6 are composite.

79, 83, 127, 131, 163, 167, 379, 383, 397, 401, 439, 443, 487, 491, 499, 503, 673, 677, 739, 743, 757, 761, 769, 773, 907, 911, 937, 941, 967, 971, 1009, 1013, 1213, 1217, 1549, 1553, 1567, 1571, 1579, 1583, 1597, 1601, 2203, 2207, 2293, 2297

Offset

1,1

Comments

List of prime numbers that occur in pairs of the form {p, p+4} after the sequential removal, from a list of all the primes, of (1) the one pair of primes of the form {p, p+1}, (2) all remaining twin prime pairs {p, p+2}.

Conjecture: the sequence has infinitely many terms and the sum of their reciprocals converges.

The second half of the conjecture is correct. The first half is true on Dickson's conjecture (because, for example, it would show that there are infinitely many k such that 210k+127 and 210k+131 are both prime). - Charles R Greathouse IV, Nov 20 2016

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Prog

(PARI) {
/* For biggest n allocatemem(max)*/
n=10^4-1;
v=vector(n, unused, 1);
for(i=2, sqrt(n),
    if(v[i],
       forstep(j=i^2, n, i, v[j]=0))
      );
v[2]=0; v[3]=0;
for(i=5, n-2,
     if(v[i]&&v[i+2],
        v[i]=0; v[i+2]=0
       )
    );
for(i=5, n-4,
     if(v[i]&&v[i+4],
        print1(i", "i+4", ");
        v[i]=0; v[i+4]=0;
       )
    )
}
(PARI) p=2; q=3; r=5; forprime(s=7, 1e3, if(r-q==4 && q-p>2 && s-r>2, print1(q", "r", ")); p=q; q=r; r=s) \\ Charles R Greathouse IV, Nov 19 2016

Crossrefs

Cf. A001097, A007510.

Sequence in context: A051326 A033399 A272639 * A033251 A015984 A235227

Adjacent sequences: A275018 A275019 A275020 * A275022 A275023 A275024

Keyword

nonn,easy

Author

Dimitris Valianatos, Nov 12 2016

Status

approved