A229256 Difference between PrimePi(10^n) and its approximation by A229255(n).

0, 0, 0, 0, 0, 10, 223, 144, -9998, -58280, 348134, 9517942, 92182430, 404027415, -2717447318, -79612186200, -983858494247, -7964818545554, -31776540093807, 289145607666924, 8243854930562789, 108476952917770938, 885519807642948390, 715407405727600672, -147909423143942345447

Offset: 1

Author

Vladimir Pletser, Sep 17 2013

Keywords

• sign
• less

Comments

A229255 provides exact values of pi(10^n) for n=1 to 5 and yields an average relative difference in absolute value of Average(Abs(A229256(n))/pi(10^n)) = 2.05820...*10^-4 for 1<=n<=25.

A229255 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), Average(Abs(A057794 (n))/pi(10^n)) =1.219...*10^-2; (2) the functions of the logarithmic integral Li(x) whether as the sequence of integer nearest to (Li(10^n)-Li(3)) (A223166) (Average(Abs(A223167(n))/pi(10^n))= 7.4969...*10^-3), or as Gauss’ approximation to pi(10^n), i.e. the sequence of integer nearest to (Li(10^n)-Li(2)) (A190802) (Average(Abs(A106313(n))/pi(10^n)) =2.0116...*10^-2), or as the sequence of integer nearest to Li(10^n) (A057752) (Average(Abs(A057752 (n))/pi(10^n)) =3.2486...*10^-2).

References

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

Links

• Vladimir Pletser, Table of n, a(n) for n = 1..25
• C. K. Caldwell, How Many Primes Are There?
• Eric Weisstein's World of Mathematics, Prime Counting Function
• Eric Weisstein's World of Mathematics, Logarithmic Integral
• Wikipedia, Prime-counting function

Crossrefs

Cf. A006880, A229255, A225137, A215663, A057793, A057794, A223166, A223167, A190802, A106313, A057752, A227693, A052435.

Formula

a(n) = A006880(n) - A229255(n).