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%I #8 Apr 18 2021 22:01:34
%S 0,0,0,1,0,-31,-35,193,0,-13318,-153006,-828603,957634,86210559,
%T 1293461717,13497122460,107995231864,586760026575,-1942949,
%U -54073500144915,-897247302459084,-9393904607181950,-54876701507521387,379565456321952448
%N Difference between pi(10^n) and nearest integer to (4*((S(n))^(n-1))) where pi(10^n) = number of primes <= 10^n (A006880) and S(n) = Sum_{i=0..2} (C(i)*(log(log(A*(B+n^(8/3)))))^(2i)) (A225137).
%C A225137 provides exactly the values of pi(10^n) for n = 1, 2, 3, 5 and 9 and yields an average relative difference in absolute value, i.e., average(abs(A225138(n))/pi(10^n)) = 7.2165...*10^-5 for 1 <= n <= 24.
%C A225137 provides a better approximation to the distribution of pi(10^n) than: (1) the Riemann function R(10^n), whether as the sequence of integers <= R(10^n) (A215663), which yields 1.453...*10^-4, or as the sequence of integers nearest to R(10^n) (A057794), which yields 0.01219...; (2) the functions of the logarithmic integral Li(x) = Integral_{t=0..x} dt/log(t), whether as the sequence of integers nearest to (Li(10^n) - Li(3)) (A223166), which yields 7.4969...x10^-3 (see A223167), or as Gauss's approximation to pi(10^n), i.e., the sequence of integers nearest to (Li(10^n) - Li(2)) (A190802) = 0.020116... (see A106313), or as the sequence of integers nearest to Li(10^n) (A057752), which yields 0.032486....
%D Jonathan Borwein, David H. Bailey, Mathematics by Experiment, A. K. Peters, 2004, p. 65 (Table 2.2).
%D John H. Conway and R. K. Guy, The Book of Numbers, Copernicus, an imprint of Springer-Verlag, NY, 1996, page 144.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeCountingFunction.html">Prime Counting Function.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RiemannPrimeNumberFormula.html">Riemann Prime Number Formula.</a>
%F a(n) = A006880(n) - A225137(n).
%Y Cf. A006880, A225137, A215663, A057794, A223166, A223167, A190802, A106313, A057752.
%K sign
%O 1,6
%A _Vladimir Pletser_, Apr 29 2013
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