A218275 a(n) is the smallest n-isolated prime, or a(n)=0 if there are no n-isolated primes.

5, 7, 11, 89, 359, 211, 1913, 2053, 1087, 1657, 4177, 2503, 7993, 6917, 4327, 11213, 5623, 24281, 54251, 17257, 31397, 62383, 85991, 25523, 37747, 35617, 259907, 143053, 188107, 181361, 369581, 1179109, 290317, 190471, 206699, 370261, 1130863, 162143

Offset: 2

Author

Vladimir Shevelev and Zak Seidov, Oct 25 2012

Keywords

• nonn

Comments

For a given n>=2, a prime p such that there is no other prime in the interval [n*prevprime(p/n), n*nextprime(p/n)] is called n-isolated.

Conjectures. 1) a(n) > 0; 2) a(n)/n is between 2 and 3 or between the smaller and larger member of a twin prime pair.

Examples

			a(5) = 89 because there are no primes except 89 in the interval [5*prevprime(89/5), 5*nextprime(89/5)] = [5*17, 5*19] = [85, 95].  And 89 is the smallest such prime - for example, if q = 37 < 89, then the interval [5*nextprime(q/5), 5*nextprime(q/5)] = [5*7,5*11] = [35,55] contains 4 primes other than 41, namely 37, 43, 47, and 53. - _Vladimir Shevelev_, Nov 04 2012.
		

Links

• Zak Seidov, Table of n, a(n) for n = 2..200
• 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

Mathematica

s = {}; Do[a = 2; b = 3; While[(p = NextPrime[k*a]) != NextPrime[k*b, -1], a = b; b = NextPrime[b]]; AppendTo[s, p], {k, 2, 40}]; s (* Zak Seidov, Nov 04 2012 *)

Formula

nextprime(a(n)/n) < nextprime(a(n))/n. For n>=5 and every prime q from the interval (3*n, a(n)), the interval (n*prevprime(q/n), n*nextprime(q/n)) contains a prime greater than q. - Vladimir Shevelev, Nov 04 2012