

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
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OFFSET

1,5


COMMENTS

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).


REFERENCES

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 SpringerVerlag, NY, 1996, page 144.


LINKS

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.


FORMULA

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


CROSSREFS

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


KEYWORD

sign


AUTHOR

Vladimir Pletser, Jul 19 2013


STATUS

approved



