

A166712


Number of primes in (n*log(n)..(n+1)*log(n+1)] semiopen intervals, n >= 1.


3



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
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OFFSET

1,2


COMMENTS

The semiopen intervals form a partition of the real line for x > 0, thus each prime appears in a unique interval.
The nth interval length is:
log(n+1/2)+1
log(n) as n goes to infinity
The nth 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 nth 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


FORMULA

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


MATHEMATICA

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


CROSSREFS

Cf. A166363, A000720.
Sequence in context: A339210 A176451 A091297 * A035183 A178101 A324831
Adjacent sequences: A166709 A166710 A166711 * A166713 A166714 A166715


KEYWORD

nonn


AUTHOR

Daniel Forgues, Oct 19 2009, Oct 23 2009


STATUS

approved



