

A215663


Floor(R(10^n))  pi(10^n), where pi(x) is the number of primes <= x, R(x) = Sum_{ k>=1 } ((mu(k)/k) * li(x^(1/k))) and li(x) is the Cauchy principal value of the integral from 0 to x of dt/log(t).


4



0, 0, 0, 3, 5, 29, 88, 96, 79, 1828, 2319, 1476, 5774, 19201, 73217, 327052, 598255, 3501366, 23884333, 4891825, 86432205, 127132665, 1033299853, 1658989720, 1834784715, 17149335456, 17535487935, 174760519828
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OFFSET

1,4


COMMENTS

In Riemann's approximation for the number of primes <= 10^n, taking Floor(R(10^n)), i.e. the greatest integer <= R(10^n), instead of the nearest integer to R(10^n), i.e. Round(R(10^n)) (see A057794), provides a better approximation to pi(10^n) for small values of n and some other values of n, i.e. Abs(a(n)) = Abs(A057794(n))1 for n = 1, 2, 8, 15. However, the approximation is worse by one unit, i.e. Abs(a(n)) = Abs(A057794(n))+1 for n = 4, 11, 13, 14, 21, 24, 25, 27, 28. The approximation is the same for the other 15 values of n <= 28. However, it yields a better average relative difference, i.e. Average(Abs(a(n))/pi(10^n)) = 1.24535…x10^4 for 1 <= n <= 28, compared to Average(Abs(A057794(n))/pi(10^n)) = 1.04526…x10^2.  Corrected and extended by Eduard Roure Perdices, Apr 16 2021


REFERENCES

John H. Conway and R. K. Guy, The Book of Numbers, Copernicus, an imprint of SpringerVerlag, NY, 1996, page 146.


LINKS



MATHEMATICA

R[x_] := Sum[N[LogIntegral[x^(1/k)]*MoebiusMu[k]/k, 36], {k, 1, 1000}]; a[n_] := Floor[R[10^n]PrimePi[10^n]]


CROSSREFS



KEYWORD

sign


AUTHOR



EXTENSIONS



STATUS

approved



