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

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A157301 Reduced numerators of the ratios of Pi(2^n+1)/Pi(2^(n)). 0

%I

%S 2,2,3,11,18,31,54,97,172,309,188,257,475,878,3271,12251,23000,4339,

%T 16405,155611,295947,564163,1077871,2063689,3957809,7603553,14630843,

%U 28192750,27200014,105097565,203280221,393615806,762939111,493402093

%N Reduced numerators of the ratios of Pi(2^n+1)/Pi(2^(n)).

%C The ratios Pi(2^n)/Pi(2^(n-1)) ~ 2. This follows directly from the Prime Number Theorem: Pi(x) ~ x/log(x). If we substitute b for 2, we have the general asymptotic Pi(b^n)/Pi(b^(n-1)) ~ b for any base b. For example, using Li(x) ~ Pi(x), Li(2^10000)/Li(2^9999) = 1.9997999711... Similarly, for b=13, Li(13^100000)/Li(13^99999) = 12.9998699994...Of course direct substitution of x=b^n in the PNT will, after some manipulation and taking limits, give us the exact limit b.

%F Pi(n) is the number of primes less than or equal to n.

%e Pi(2^12)/Pi(2^11) = 564/309 = 188/103. So 188 is in the sequence.

%o (PARI) /* Copy and paste the table in A007053 to a text file say, c:\work\test.txt.

%o Edit out the index leaving only a left wall of values. Start a new gp session. Read the file into gp: gp > \r c:/work/test.txt. This fills the %1 to %76 pari variables with successive primes <= 2^n

%o */

%o for(j=2,75,x=eval(concat("%",j+1));

%o y=eval(concat("%",j));z=numerator(x/y);print1(z","))

%Y Cf. A007053

%K nonn

%O 2,1

%A _Cino Hilliard_, Feb 26 2009

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.

Last modified December 7 09:15 EST 2021. Contains 349574 sequences. (Running on oeis4.)