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

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, 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, 1, 0, 1, 2, 1, 0, 0, 1, 1, 0, 2, 1, 1, 0, 1, 1, 2, 0, 1

Offset: 1

Author

Daniel Forgues, Oct 19 2009, Oct 23 2009

Keywords

• nonn

Comments

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

The n-th interval length is:

log(n+1/2)+1

log(n) as n goes to infinity

The n-th interval prime density is:

1/[log(n+1/2)+log(log(n+1/2))]

1/log(n) as n goes to infinity

The expected number of primes for n-th interval is:

[log(n+1/2)+1] / [log(n+1/2)+log(log(n+1/2))]

1 as n goes to infinity (for expected number of primes per interval)

The expected number of primes per interval is asymptotic to 1.

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.)

The average of the actual number of primes per interval for all intervals from 1 to n is asymptotic to 1.

The sequence first attains k = 0, 1, 2,... at n = 1, 3, 2, 234, 3843, 71221,... - T. D. Noe, Oct 15 2012

Links

• Daniel Forgues, Table of n, a(n) for n=1..10769

Crossrefs

Cf. A166363, A000720.

Programs

Mathematica

Table[PrimePi[(n+1)*Log[n+1]] - PrimePi[n*Log[n]], {n, 100}] (* T. D. Noe, Oct 15 2012 *)

Formula

a(n) = pi((n+1)*log(n+1)) - pi(n*log(n)) since the intervals are semi-open properly.