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A166712 Number of primes in (n*log(n)..(n+1)*log(n+1)] semi-open intervals, n >= 1. 3

%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)] semi-open intervals, n >= 1.

%C The semi-open intervals form a partition of the real line for x > 0, thus each prime appears in a unique interval.

%C The n-th interval length is:

%C log(n+1/2)+1

%C log(n) as n goes to infinity

%C The n-th 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 n-th 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 semi-open 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

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Last modified May 7 02:20 EDT 2021. Contains 343636 sequences. (Running on oeis4.)