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Search: id:A157301
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| A157301 |
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Reduced numerators of the ratios of Pi(2^n+1)/Pi(2^(n)). |
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+0
1
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| 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; listen)
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OFFSET
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2,1
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COMMENT
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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 asymtotic 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.
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FORMULA
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Pi(n) is the number of primes less than or equal to n.
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EXAMPLE
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Pi(2^12)/Pi(2^11) = 564/309 = 188/103. So 188 is in the sequence.
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PROGRAM
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(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", "))
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CROSSREFS
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Cf. A007053
Sequence in context: A019143 A084650 A067919 this_sequence A143931 A143933 A075095
Adjacent sequences: A157298 A157299 A157300 this_sequence A157302 A157303 A157304
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KEYWORD
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nonn
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AUTHOR
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Cino Hilliard (hillcino368(AT)hotmail.com), Feb 26 2009
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