login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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