1,1

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.

(PARI) g(n) = for(x=1, n, y=floor(2^x/log(2^x)); print1(y", "))

Sequence in context: A025591 A028409 A080553 this_sequence A130377 A049873 A127180

Adjacent sequences: A141599 A141600 A141601 this_sequence A141603 A141604 A141605

nonn,uned

Cino Hilliard (hillcino368(AT)hotmail.com), Aug 21 2008