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%I #11 Mar 08 2023 09:42:45
%S 6,26,168,1217,9511,78029,661458,5740303,50701541,454011970,
%T 4110416300,37550193649,345618860220,3201414635780,29816233849000,
%U 279007258230819,2621647966812030,24723998785919975,233922961602470390
%N 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.
%C 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.
%C For little googol = 2^100 the formula gives 18556039405581571438895944827, while Riemann's R(x) = 18560140176092446446103729058.
%C 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.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeNumberTheorem.html">Prime Number Theorem</a>.
%e For x = 10^3 a(3) = 168.
%o (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)","))
%K nonn
%O 1,1
%A _Cino Hilliard_, Aug 28 2008
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