%I #21 Dec 08 2016 10:23:13
%S 47,53,251,257,331,337,353,359,367,373,541,547,557,563,587,593,607,
%T 613,647,653,727,733,947,953,977,983,991,997,1097,1103,1117,1123,1181,
%U 1187,1361,1367,1433,1439,1493,1499,1657,1663,1741,1747,1753
%N Primes not in A001097 and not in A275021 that occur in pairs of the form {p, p+6}.
%C List of prime numbers that occur in pairs of the form {p, p+6} 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}, and (3) all remaining pairs of the form {p, p+4}.
%C Conjecture: The sequence has infinitely many terms and the sum of their reciprocals converges.
%o (PARI) {
%o /* For biggest n allocatemem(max)*/
%o n=10^4-1;
%o v=vector(n, unused, 1);
%o for(i=2, sqrt(n),
%o if(v[i],
%o forstep(j=i^2, n, i, v[j]=0))
%o );
%o v[2]=0; v[3]=0;
%o for(i=5, n-2,
%o if(v[i]&&v[i+2],
%o v[i]=0; v[i+2]=0
%o )
%o );
%o for(i=5, n-4,
%o if(v[i]&&v[i+4],
%o v[i]=0; v[i+4]=0;
%o )
%o );
%o for(i=5, n-6,
%o if(v[i]&&v[i+6],
%o print1(i", "i+6", ");
%o v[i]=0; v[i+6]=0;
%o )
%o )
%o }
%Y Cf. A001097, A275021.
%K nonn,more
%O 1,1
%A _Dimitris Valianatos_, Nov 12 2016