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 A166712 Number of primes in (n*log(n)..(n+1)*log(n+1)] semi-open 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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 FORMULA a(n) = pi((n+1)*log(n+1)) - pi(n*log(n)) since the intervals are semi-open 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

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Last modified May 16 22:04 EDT 2021. Contains 343955 sequences. (Running on oeis4.)