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 A225138 Difference between pi(10^n) and nearest integer to (4*((S(n))^(n-1))) where pi(10^n) = number of primes <= 10^n (A006880) and S(n) = Sum_{i=0..2} (C(i)*(log(log(A*(B+n^(8/3)))))^(2i)) (A225137). 2
 0, 0, 0, 1, 0, -31, -35, 193, 0, -13318, -153006, -828603, 957634, 86210559, 1293461717, 13497122460, 107995231864, 586760026575, -1942949, -54073500144915, -897247302459084, -9393904607181950, -54876701507521387, 379565456321952448 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS A225137 provides exactly the values of pi(10^n) for n = 1, 2, 3, 5 and 9 and yields an average relative difference in absolute value, i.e., average(abs(A225138(n))/pi(10^n)) = 7.2165...*10^-5 for 1 <= n <= 24. A225137 provides a better approximation to the distribution of pi(10^n) than: (1) the Riemann function R(10^n), whether as the sequence of integers <= R(10^n) (A215663), which yields 1.453...*10^-4, or 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 7.4969...x10^-3 (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) = 0.020116... (see A106313), or as the sequence of integers nearest to Li(10^n) (A057752), which yields 0.032486.... 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 Springer-Verlag, NY, 1996, page 144. LINKS Table of n, a(n) for n=1..24. Eric Weisstein's World of Mathematics, Prime Counting Function. Eric Weisstein's World of Mathematics, Riemann Prime Number Formula. FORMULA a(n) = A006880(n) - A225137(n). CROSSREFS Cf. A006880, A225137, A215663, A057794, A223166, A223167, A190802, A106313, A057752. Sequence in context: A338068 A036228 A299455 * A300093 A109293 A053684 Adjacent sequences: A225135 A225136 A225137 * A225139 A225140 A225141 KEYWORD sign AUTHOR Vladimir Pletser, Apr 29 2013 STATUS approved

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Last modified June 10 10:09 EDT 2023. Contains 363199 sequences. (Running on oeis4.)