A103649 Number of primes less than 10^n using the x-th root approximation formula 1/(x^(1/x) - 1/x - 1) where x = 10^n.

6, 26, 168, 1217, 9511, 78029, 661458, 5740303, 50701541, 454011970, 4110416300, 37550193649, 345618860220, 3201414635780, 29816233849000, 279007258230819, 2621647966812030, 24723998785919975, 233922961602470390

Offset

1,1

Comments

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, while 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.

Links

Table of n, a(n) for n=1..19.

Eric Weisstein's World of Mathematics, Prime Number Theorem.

Example

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

Prog

(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)", "))

Crossrefs

Sequence in context: A100308 A352694 A049040 * A053946 A211946 A027283

Adjacent sequences: A103646 A103647 A103648 * A103650 A103651 A103652

Keyword

nonn

Author

Cino Hilliard, Aug 28 2008

Status

approved