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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). 2
0, 0, 0, 0, -3, -29, 171, 2325, 13809, 33409, -443988, -8663889, -99916944, -927360109, -7318034084, -47993181878, -223530657736, 810207694, 16558446000251, 257071298610935, 2657469557986545, 18804132783879606, 24113768300809752, -2232929440358147845, -54971510676262602742 (list; graph; refs; listen; history; text; internal format)



The sequence (A227693) provides exactly the values of pi(10^n) for n=1 to 4 and yields an average relative difference in absolute value, i.e. <ARD(A227694)> =Average(Abs(A227694 (n))/pi(10^n)) = 1.58269…x10^-4 for 1<=n<=25.

The sequence (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) (<ARD(A057794)> =1.219…x10^-2); (2) the functions of the logarithmic integral Li(x) = integral(0..x, dt/log(t)) whether as the sequence of integer nearest to (Li(10^n)-Li(3)) (A223166) (<ARD(A223167)>= 7.4969…x10^-3), or as Gauss’ approximation to pi(10^n), i.e. the sequence of integer nearest to (Li(10^n)-Li(2)) (A190802) (<ARD(A106313)> =2.0116…x10^-2), or as the sequence of integer nearest to Li(10^n) (A057752) (<ARD(A057752)> =3.2486…x10^-2).


Jonathan Borwein, David H. Bailey, Mathematics by Experiment, A. K. Peters, 2004, p. 65 (Table 2.2).

John H. Conway and R. K. Guy, The Book of Numbers, Copernicus, an imprint of Springer-Verlag, NY, 1996, page 144.


Table of n, a(n) for n=1..25.

Eric Weisstein's World of Mathematics, Prime Counting Function.

Eric Weisstein's World of Mathematics, Riemann Prime Number Formula.


a(n) = A006880(n) - A227693(n).


Cf. A006880, A225137, A215663, A057794, A223166, A223167, A190802, A106313, A057752, A227693.

Sequence in context: A096028 A137786 A112498 * A118584 A126185 A083092

Adjacent sequences:  A227691 A227692 A227693 * A227695 A227696 A227697




Vladimir Pletser, Jul 19 2013



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Last modified August 8 05:50 EDT 2020. Contains 336290 sequences. (Running on oeis4.)