

A103649


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


0



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

1,1


COMMENTS

This formula was derived from the xth 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.
E. Weisstein 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: A187458 A100308 A049040 * A053946 A211946 A027283
Adjacent sequences: A103646 A103647 A103648 * A103650 A103651 A103652


KEYWORD

nonn


AUTHOR

Cino Hilliard, Aug 28 2008


STATUS

approved



