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

2,1

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.

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

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

Example

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.

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 *)

Crossrefs

Cf. A166251, A217561, A217566, A217577.

Sequence in context: A071781 A091509 A027728 * A086140 A104387 A133761

Adjacent sequences: A218272 A218273 A218274 * A218276 A218277 A218278

Keyword

nonn

Author

Vladimir Shevelev and Zak Seidov, Oct 25 2012

Status

approved