cp's OEIS Frontend

A166712 Number of primes in (n*log(n)..(n+1)*log(n+1)] semi-open intervals, n >= 1.

%I A166712 #7 Oct 15 2012 15:05:46
%S A166712 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 A166712 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 A166712 1,0,1,2,1,0,0,1,1,0,2,1,1,0,1,1,2,0,1
%N A166712 Number of primes in (n*log(n)..(n+1)*log(n+1)] semi-open intervals, n >= 1.
%C A166712 The semi-open intervals form a partition of the real line for x > 0, thus each prime appears in a unique interval.
%C A166712 The n-th interval length is:
%C A166712 log(n+1/2)+1
%C A166712 log(n) as n goes to infinity
%C A166712 The n-th interval prime density is:
%C A166712 1/[log(n+1/2)+log(log(n+1/2))]
%C A166712 1/log(n) as n goes to infinity
%C A166712 The expected number of primes for n-th interval is:
%C A166712 [log(n+1/2)+1] / [log(n+1/2)+log(log(n+1/2))]
%C A166712 1 as n goes to infinity (for expected number of primes per interval)
%C A166712 The expected number of primes per interval is asymptotic to 1.
%C A166712 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 A166712 The average of the actual number of primes per interval for all intervals from 1 to n is asymptotic to 1.
%C A166712 The sequence first attains k = 0, 1, 2,... at n = 1, 3, 2, 234, 3843, 71221,... - _T. D. Noe_, Oct 15 2012
%H A166712 Daniel Forgues, <a href="/A166712/b166712.txt">Table of n, a(n) for n=1..10769</a>
%F A166712 a(n) = pi((n+1)*log(n+1)) - pi(n*log(n)) since the intervals are semi-open properly.
%t A166712 Table[PrimePi[(n+1)*Log[n+1]] - PrimePi[n*Log[n]], {n, 100}] (* _T. D. Noe_, Oct 15 2012 *)
%Y A166712 Cf. A166363, A000720.
%K A166712 nonn
%O A166712 1,2
%A A166712 _Daniel Forgues_, Oct 19 2009, Oct 23 2009