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

2, 2, 3, 11, 18, 31, 54, 97, 172, 309, 188, 257, 475, 878, 3271, 12251, 23000, 4339, 16405, 155611, 295947, 564163, 1077871, 2063689, 3957809, 7603553, 14630843, 28192750, 27200014, 105097565, 203280221, 393615806, 762939111, 493402093

Offset

2,1

Comments

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.

Links

Table of n, a(n) for n=2..35.

Formula

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

Example

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

Mathematica

Table[Numerator[PrimePi[2^(n+1)]/PrimePi[2^n]], {n, 40}] (* Harvey P. Dale, Sep 22 2023 *)

Prog

(PARI) /* Copy and paste the table in A007053 to a text file say, c:\work\test.txt.
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
*/
for(j=2, 75, x=eval(concat("%", j+1));
y=eval(concat("%", j)); z=numerator(x/y); print1(z", "))

Crossrefs

Cf. A007053

Sequence in context: A019143 A084650 A067919 * A143931 A143933 A284708

Adjacent sequences: A157298 A157299 A157300 * A157302 A157303 A157304

Keyword

nonn

Author

Cino Hilliard, Feb 26 2009

Extensions

Definition clarified by Harvey P. Dale, Sep 22 2023

Status

approved