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%I #55 Nov 02 2013 21:57:09
%S 2,2,2,17,59,29,239,227,107,149,347,191,569,461,269,659,311,1277,2711,
%T 821,1427,2711,3581,1019,1451,1319,9281,4931,6269,5849,11549,35729,
%U 8537,5441,5741,10007,29759
%N a(n) is the smallest prime(m) such that the interval (prime(m)*n, prime(m+1)*n) contains exactly one prime.
%C Conjecture: In the supposition that there are infinitely many twin primes, for n>=5 all terms are in A001359 (lesser of twin primes).
%C Note that a unique prime which is contained in an interval of the form (prime(m)*n, prime(m+1)*n) is called n-isolated (see author's link, where a heuristic proof is given that the number of n-isolated primes<=x approaches e^{-2(n-1)}x/log(x) as x goes to infinity (cf. Conjecture 25, Remark 26 and formula (47)). One can easily prove that a(n) is not bounded.
%C This conjecture seems hard, since it's not obvious how to find an upper bound for a(n) (see Conjecture 42 in the Shevelev link). - _Charles R Greathouse IV_, Jan 02 2013
%H Charles R Greathouse IV, <a href="/A195871/b195871.txt">Table of n, a(n) for n = 2..100</a>
%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
%H J. Sondow, J. W. Nicholson, and T. D. Noe, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Noe/noe12.html">Ramanujan Primes: Bounds, Runs, Twins, and Gaps</a>, J. Integer Seq. 14 (2011), Article 11.6.2
%e Let n=5, and consider intervals of the form (5*prime(m), 5*prime(m+1)).
%e For 2, 3, 5, ..., the intervals (10,15), (15,25), (25,35), (35,55), (55,65), (65,85), (85,95)... contain 2, 3, 2, 5, 2, 4, 1,... primes. Hence the smallest such prime is 17.
%o (PARI) a(n)=my(p=2,t);forprime(q=3,,t=0;for(i=p*n+1,q*n-1,if(isprime(i)&&t++>1,break));if(t==1,return(p));p=q) \\ _Charles R Greathouse IV_, Jan 02 2013
%Y Cf. A166251, A217561, A217566, A217577.
%K nonn
%O 2,1
%A _Vladimir Shevelev_, Jan 02 2013