%I
%S 0,2,1,1,0,2,0,2,1,0,1,1,0,1,2,1,0,1,1,1,1,1,1,1,1,0,1,1,1,1,2,1,0,0,
%T 1,1,1,1,0,2,0,1,1,1,1,1,1,0,2,2,0,0,1,0,1,2,1,1,1,0,1,1,1,2,0,2,1,0,
%U 1,0,1,2,1,0,0,1,1,0,2,1,1,0,1,1,2,0,1
%N Number of primes in (n*log(n)..(n+1)*log(n+1)] semiopen intervals, n >= 1.
%C The semiopen intervals form a partition of the real line for x > 0, thus each prime appears in a unique interval.
%C The nth interval length is:
%C log(n+1/2)+1
%C log(n) as n goes to infinity
%C The nth interval prime density is:
%C 1/[log(n+1/2)+log(log(n+1/2))]
%C 1/log(n) as n goes to infinity
%C The expected number of primes for nth interval is:
%C [log(n+1/2)+1] / [log(n+1/2)+log(log(n+1/2))]
%C 1 as n goes to infinity (for expected number of primes per interval)
%C The expected number of primes per interval is asymptotic to 1.
%C The actual number of primes per interval is not asymptotic to 1 since it does not actually converge but just keeps on hitting cardinals on and around 1 (mostly 0, 1 and 2.)
%C The average of the actual number of primes per interval for all intervals from 1 to n is asymptotic to 1.
%C The sequence first attains k = 0, 1, 2,... at n = 1, 3, 2, 234, 3843, 71221,...  _T. D. Noe_, Oct 15 2012
%H Daniel Forgues, <a href="/A166712/b166712.txt">Table of n, a(n) for n=1..10769</a>
%F a(n) = pi((n+1)*log(n+1))  pi(n*log(n)) since the intervals are semiopen properly.
%t Table[PrimePi[(n+1)*Log[n+1]]  PrimePi[n*Log[n]], {n, 100}] (* _T. D. Noe_, Oct 15 2012 *)
%Y Cf. A166363, A000720.
%K nonn
%O 1,2
%A _Daniel Forgues_, Oct 19 2009, Oct 23 2009
