A195871 a(n) is the smallest prime(m) such that the interval (prime(m)*n, prime(m+1)*n) contains exactly one prime.

2, 2, 2, 17, 59, 29, 239, 227, 107, 149, 347, 191, 569, 461, 269, 659, 311, 1277, 2711, 821, 1427, 2711, 3581, 1019, 1451, 1319, 9281, 4931, 6269, 5849, 11549, 35729, 8537, 5441, 5741, 10007, 29759

Offset: 2

Author

Vladimir Shevelev, Jan 02 2013

Keywords

• nonn

Comments

Conjecture: In the supposition that there are infinitely many twin primes, for n>=5 all terms are in A001359 (lesser of twin primes).

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.

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

Examples

			Let n=5, and consider intervals of the form (5*prime(m), 5*prime(m+1)).
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.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 2..100
• V. Shevelev, Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes, Journal of Integer Sequences, Vol. 15 (2012), Article 12.5.4
• J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011), Article 11.6.2

Crossrefs

Cf. A166251, A217561, A217566, A217577.

Programs

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