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 A157301 Reduced numerators of the ratios of Pi(2^n+1)/Pi(2^(n)). 0
 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 (list; graph; refs; listen; history; text; internal format)
 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 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. 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 STATUS approved

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Last modified October 23 08:40 EDT 2021. Contains 348211 sequences. (Running on oeis4.)