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A227694 Difference between pi(10^n) and nearest integer to (F[2n+1](S(n)))^2 where pi(10^n) = number of primes <= 10^n (A006880), F[2n+1](x) are Fibonacci polynomials of odd indices [2n+1] and S(n) = Sum_{i=0..2} (C(i)*(log(log(A*(B+n^2))))^(2i)) (see A227693).

%I A227694 #13 Feb 16 2025 08:33:20
%S A227694 0,0,0,0,-3,-29,171,2325,13809,33409,-443988,-8663889,-99916944,
%T A227694 -927360109,-7318034084,-47993181878,-223530657736,810207694,
%U A227694 16558446000251,257071298610935,2657469557986545,18804132783879606,24113768300809752,-2232929440358147845,-54971510676262602742
%N A227694 Difference between pi(10^n) and nearest integer to (F[2n+1](S(n)))^2 where pi(10^n) = number of primes <= 10^n (A006880), F[2n+1](x) are Fibonacci polynomials of odd indices [2n+1] and S(n) = Sum_{i=0..2} (C(i)*(log(log(A*(B+n^2))))^(2i)) (see A227693).
%C A227694 A227693 provides exactly the values of pi(10^n) for n = 1 to 4 and yields an average relative difference in absolute value, average(abs(A227694(n))/pi(10^n)) = 1.58269...*10^-4 for 1 <= n <= 25.
%C A227694 A227693 provides a better approximation to the distribution of pi(10^n) than: (1) the Riemann function R(10^n) 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 0.0074969... (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), which yields 0.020116... (see A106313), or as the sequence of integer nearest to Li(10^n) (A057752), which yields 0.032486....
%D A227694 Jonathan Borwein, David H. Bailey, Mathematics by Experiment, A. K. Peters, 2004, p. 65 (Table 2.2).
%D A227694 John H. Conway and R. K. Guy, The Book of Numbers, Copernicus, an imprint of Springer-Verlag, NY, 1996, page 144.
%H A227694 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeCountingFunction.html">Prime Counting Function.</a>
%H A227694 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RiemannPrimeNumberFormula.html">Riemann Prime Number Formula.</a>
%F A227694 a(n) = A006880(n) - A227693(n).
%Y A227694 Cf. A006880, A225137, A215663, A057794, A223166, A223167, A190802, A106313, A057752, A227693.
%K A227694 sign
%O A227694 1,5
%A A227694 _Vladimir Pletser_, Jul 19 2013