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A141602
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Integer part of 2^n/log(2^n).
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0
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2, 2, 3, 5, 9, 15, 26, 46, 82, 147, 268, 492, 909, 1688, 3151, 5909, 11123, 21010, 39809, 75638, 144073, 275050, 526182, 1008516, 1936352, 3723754, 7171675, 13831089, 26708310, 51636066, 99940774, 193635250, 375535031, 728979766, 1416303547
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| 2^n/log(2^n) is an approximation to the number of primes < 2^n. A closely
related function is x/(x^(1/x) - 1) when x is a power of 2. In general, for
any x the limit of x/log(x) - x/(x^(1/x)-1) -> 1/2 as x -> infinity. The
interesting thing about this relationship for powers of 2 is that we can
easily approximate the number of primes < x = 2^n simply by repeated square
roots to evaluate x^(1/x) in x/(x^(1/x)-1). You can do this on most
calculators but only for small x because of rounding. So we have it,
x/log(x) ~ x/(x^(1/x)-1) + 1/2 ~ Pi(x). For x = 2^n, we need not take logs
rather just square roots to compute values for the PNT. It remains to
prove this experimental result analytically.
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FORMULA
| a(n)=A050500(2^n)=floor(2^n*A007525/n) >= A000799(n). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 05 2009]
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PROG
| (PARI) g(n) = for(x=1, n, y=floor(2^x/log(2^x)); print1(y", "))
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CROSSREFS
| Sequence in context: A028409 A183559 A080553 * A153931 A130377 A153900
Adjacent sequences: A141599 A141600 A141601 * A141603 A141604 A141605
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KEYWORD
| nonn,uned
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AUTHOR
| Cino Hilliard (hillcino368(AT)hotmail.com), Aug 21 2008
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