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A195871 a(n) is the smallest prime(m) such that the interval (prime(m)*n, prime(m+1)*n) contains exactly one prime. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

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

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

EXAMPLE

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.

PROG

(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

CROSSREFS

Cf. A166251, A217561, A217566, A217577.

Sequence in context: A068218 A098919 A161748 * A214756 A079007 A064215

Adjacent sequences:  A195868 A195869 A195870 * A195872 A195873 A195874

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Jan 02 2013

STATUS

approved

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Last modified December 10 23:29 EST 2019. Contains 329910 sequences. (Running on oeis4.)