%I
%S 2,8,13,21,32,38,46,60,85,74,102,111
%N Smallest k such that there are exactly n primes between k*(k1) and k^2 and exactly n primes between k^2 and k*(k+1).
%C Does k exist for every n?
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Oppermann%27s_conjecture">Oppermann's conjecture</a>.
%e For n = 6: 38*(381) = 1406, 38^2 = 1444 and 38*(38+1) = 1482. A000720(1444)  A000720(1406) = 6 and A000720(1482)  A000720(1444) = 6. Since 38 is the smallest k where the number of primes in both intervals is 6, a(6) = 38.
%t Table[SelectFirst[Range[10^3], And[PrimePi[#^2]  PrimePi[# (#  1)] == n, PrimePi[# (# + 1)]  PrimePi[#^2] == n] &], {n, 30}] /. k_ /; MissingQ@ k > 0 (* _Michael De Vlieger_, Apr 09 2016, Version 10.2 *)
%o (PARI) a(n) = my(k=1); while((primepi(k^2)primepi(k*(k1)))!=n  (primepi(k*(k+1))primepi(k^2))!=n, k++); k
%K nonn,more
%O 1,1
%A _Felix FrÃ¶hlich_, Apr 07 2016
