A218275 - OEIS

%I #42 Jun 20 2013 03:19:28

%S 5,7,11,89,359,211,1913,2053,1087,1657,4177,2503,7993,6917,4327,11213,

%T 5623,24281,54251,17257,31397,62383,85991,25523,37747,35617,259907,

%U 143053,188107,181361,369581,1179109,290317,190471,206699,370261,1130863,162143

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

%C 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.

%C 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.

%H Zak Seidov, <a href="/A218275/b218275.txt">Table of n, a(n) for n = 2..200</a>

%H V. Shevelev, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Shevelev/shevelev19.html">Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes</a>, Journal of Integer Sequences, Vol. 15 (2012), Article 12.5.4

%H J. Sondow, J. W. Nicholson, and T. D. Noe, <a href="http://arxiv.org/abs/1105.2249"> Ramanujan Primes: Bounds, Runs, Twins, and Gaps</a>, J. Integer Seq. 14 (2011) Article 11.6.2

%F 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

%e 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.

%t 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 *)

%Y Cf. A166251, A217561, A217566, A217577.

%K nonn

%O 2,1

%A _Vladimir Shevelev_ and _Zak Seidov_, Oct 25 2012