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%I #56 Dec 16 2018 13:03:36
%S 5,5,8,10,2,12,7,4,37,23,5,51,3,6,34,23,5,57,9,22,49,66,64,54,5,56,43,
%T 28,46,116,56,232,92,170,65,206,181,379,170,511,190,416,187,448,89,
%U 143,200,159,434,670,145,1081,213,1011,680,77
%N The index of the smallest n-isolated prime p such that p/n is not between 2 and 3 and not between the smaller and greater primes of a twin prime pair, or 0 if no such p exists.
%C For n>=2, a prime p is called n-isolated (cf. A166251 and the Shevelev link, Section 10) if there is no other prime in the interval (n*prevprime(p/n), n*nextprime(p/n)).
%C In particular, if a(n)=1, then the smallest n-isolated prime divided by n is not between 2 and 3 and not between the smaller and greater primes of a twin prime pair.
%C Suppose that for every n there exist infinitely many n-isolated primes. Then if there exists n_0 such that a(n_0)=0, there are infinitely many twin primes. On the other hand, one can prove that the smallest n-isolated prime divided by n tends to infinity as n goes to infinity. Therefore, if there is not an N such that, for all n >= N, a(n)=1, then we also conclude that there are infinitely many twin primes.
%C Conjecture: all a(n) >= 2.
%H V. Shevelev, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Shevelev/shevelev19.html">Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes</a>, Journal of Integer Sequences, Vol. 15 (2012), Article 12.5.4
%e Let n=2. The 2-isolated primes are in A166251.
%e The first 2-isolated prime is 5, and 5/2 is between 2 and 3.
%e The second 2-isolated prime is 7, and 7/2 is between 3 and 5.
%e The third 2-isolated prime is 23, and 23/2 is between 11 and 13.
%e The fourth 2-isolated prime is 37, and 37/2 is between 17 and 19.
%e The fifth 2-isolated prime is 79, and 79/2 is between 37 and 41. Since (37,41) is not (2,3) and is not a twin prime pair, a(2)=5
%o (PARI) isoki(p, n) = (p==nextprime(n*precprime(p\n))) && (p==precprime(n*nextprime(p/n))); \\ A166251
%o nextp(p, n) = while(! isoki(p, n), p = nextprime(p+1)); p;
%o isokp(p, n) = {my(diff = nextprime(p/n) - precprime(p/n)); if ((diff == 1) || (diff == 2), return (0)); return (1);}
%o a(n) = {my(p = nextp(2, n), nb = 1); while (! isokp(p, n), p = nextp(nextprime(p+1), n); nb++;); nb;} \\ _Michel Marcus_, Dec 16 2018
%Y Cf. A166251, A217561, A217566.
%K nonn
%O 2,1
%A _Vladimir Shevelev_, Oct 26 2012
%E a(6)-a(38) were calculated by _Zak Seidov_, Oct 28 2012
%E More terms from _Michel Marcus_, Dec 16 2018
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