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A271383 Smallest k such that there are exactly n primes between k*(k-1) and k^2 and exactly n primes between k^2 and k*(k+1). 0

%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*(k-1) 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*(38-1) = 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*(k-1)))!=n || (primepi(k*(k+1))-primepi(k^2))!=n, k++); k

%K nonn,more

%O 1,1

%A _Felix Fröhlich_, Apr 07 2016

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Last modified October 18 14:44 EDT 2019. Contains 328161 sequences. (Running on oeis4.)