1,1

This formula was derived from the x-th root formula 1/(x^(1/x) - 1)+ 1/2 and

the well known approximation Pi(x) ~ x/(log(x) - 1). If x = 2^n, the formula

can be evaluated by repeated square roots avoiding logs.

For little googol = 2^100 the formula gives

18556039405581571438895944827. Riemann's R(x) =

18560140176092446446103729058. The formula is much more accurate than x/log(x)

and for small x, Legendre's constant 1.08366 can be used for the 1/x term as

1.08366/x. This is more accurate for small x. However, for large x, the more

noble formula 1/(x^(1/x) - 1/x - 1) is superior.

E. Weisstein Prime Number Theorem

For x = 10^3 a(3) = 168.

(PARI) /* b = 10 in this sequence */ g(n, b) = for(j=1, n, x=b^j; y=1/(x^(1/x) - 1/x -1); print1(floor(y)", "))

Sequence in context: A144037 A100308 A049040 this_sequence A053946 A027283 A009639

Adjacent sequences: A103646 A103647 A103648 this_sequence A103650 A103651 A103652

nonn

Cino Hilliard (hillcino368(AT)hotmail.com), Aug 28 2008