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Difference between PrimePi(10^n) and its approximation by A229255(n).
2

%I #29 Apr 03 2023 10:36:13

%S 0,0,0,0,0,10,223,144,-9998,-58280,348134,9517942,92182430,404027415,

%T -2717447318,-79612186200,-983858494247,-7964818545554,

%U -31776540093807,289145607666924,8243854930562789,108476952917770938,885519807642948390,715407405727600672,-147909423143942345447

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

%C 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.

%C 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).

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

%H Vladimir Pletser, <a href="/A229256/b229256.txt">Table of n, a(n) for n = 1..25</a>

%H C. K. Caldwell, <a href="https://t5k.org/howmany.shtml">How Many Primes Are There?</a>

%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/LogarithmicIntegral.html">Logarithmic Integral</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Prime-counting_function">Prime-counting function</a>

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

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

%K sign,less

%O 1,6

%A _Vladimir Pletser_, Sep 17 2013