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
KEYWORD
nonn
AUTHOR
Vladimir Shevelev and Zak Seidov, Oct 25 2012
STATUS
approved