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A157301 Reduced numerators of the ratios of Pi(2^n+1)/Pi(2^(n)). 0


%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

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