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%I #39 Dec 08 2016 10:22:43
%S 79,83,127,131,163,167,379,383,397,401,439,443,487,491,499,503,673,
%T 677,739,743,757,761,769,773,907,911,937,941,967,971,1009,1013,1213,
%U 1217,1549,1553,1567,1571,1579,1583,1597,1601,2203,2207,2293,2297
%N Pairs of primes p, p+4 such that p-2 and p+6 are composite.
%C 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}.
%C Conjecture: the sequence has infinitely many terms and the sum of their reciprocals converges.
%C 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
%H Charles R Greathouse IV, <a href="/A275021/b275021.txt">Table of n, a(n) for n = 1..10000</a>
%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 print1(i", "i+4", ");
%o v[i]=0;v[i+4]=0;
%o )
%o )
%o }
%o (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
%Y Cf. A001097, A007510.
%K nonn,easy
%O 1,1
%A _Dimitris Valianatos_, Nov 12 2016
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