A215663 -id:A215663 - cp's OEIS frontend

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).

0, 0, 0, 1, 0, -31, -35, 193, 0, -13318, -153006, -828603, 957634, 86210559, 1293461717, 13497122460, 107995231864, 586760026575, -1942949, -54073500144915, -897247302459084, -9393904607181950, -54876701507521387, 379565456321952448

Offset: 1

Vladimir Pletser, Apr 29 2013

sign

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....

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.

Eric Weisstein's World of Mathematics, Prime Counting Function.
Eric Weisstein's World of Mathematics, Riemann Prime Number Formula.

Cf. A006880, A225137, A215663, A057794, A223166, A223167, A190802, A106313, A057752.

a(n) = A006880(n) - A225137(n).

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).

Original entry on oeis.org

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

Offset: 1

Vladimir Pletser, Jul 19 2013

sign

A227693 provides exactly the values of pi(10^n) for n = 1 to 4 and yields an average relative difference in absolute value, average(abs(A227694(n))/pi(10^n)) = 1.58269...*10^-4 for 1 <= n <= 25.

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), 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 0.0074969... (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), which yields 0.020116... (see A106313), or as the sequence of integer nearest to Li(10^n) (A057752), which yields 0.032486....

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.

Eric Weisstein's World of Mathematics, Prime Counting Function.
Eric Weisstein's World of Mathematics, Riemann Prime Number Formula.

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

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

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

Original entry on oeis.org

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

Vladimir Pletser, Sep 17 2013

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).

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

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

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

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

A353055 Successive records of function f(x) = log(abs(pi(x) - R(x)))/log(x) where pi(x) is the number of primes <= x and R(x) is Riemann's prime counting function.

Original entry on oeis.org

2, 4, 7, 10, 19, 47, 58, 73, 109, 113, 1109, 1123, 1129, 1307, 1321, 1327, 1418, 1419, 1420, 1421, 1422, 5379, 5380, 7449, 7450, 10343, 11676, 11761, 11762, 11763, 11764, 11765, 11766, 11767, 11768, 11769, 11770, 11771, 11772, 11773, 11774, 11775, 11776, 19360, 19361, 19362, 19363, 19364, 19365, 19366, 19367, 19368, 19369, 19370, 19371, 19372

Offset: 1

Artur Jasinski, Apr 20 2022

nonn

Two possibilities:
(1) this sequence is finite;
(2) this sequence is infinite.
In case (1) there exists a maximal integer x_max such that J = f(x_max) = log(abs(pi(x_max) - R(x_max)))/log(x_max).
In case (2) there exists a real constant J such that lim_{x->oo} f(x) = J.
Then for every positive integer x, abs((R(x) - pi(x))/x^J) <= 1.
According to actual computations biggest x = 1090696 with log(-85020 + R(1090696))/log(1090696) = 0.27835121240340474... and no more new terms up to x 3000000. Follow this:
0.27835121240340474... <= J.
J < 1/2 = limit((log(x) - 2*log((8*Pi)/log(x)))/(2*log(x)), x -> infinity) proof follow Lowell Schoenfeld 1976 proof on upper limit of Chebyshev function psi(x).
Constant J can be used to measure best proved upper limits of asymptotical behavior of function pi(x) when x->infinity. If J is smaller upper bound is better.

			x      f(x)                comment
1     -infinity            a(1)
2     -0.8862754573970588  a(2)
3     -4.883591467412115   removed because f(3) < f(2)
4     -0.614424415865155   a(3)
5     -1.0695141714266385  removed because f(5) < f(4)
...    ...

Lowell Schoenfeld, Sharper Bounds for the Chebyshev Functions theta(x) and psi(x). II, Mathematics of Computation Vol. 30, No. 134 (Apr., 1976), pp. 337-360.

Cf. A057793, A057794, A215663, A216709, A226473.

gg = {1}; imax = -1000; Do[
kk = Log[Abs[PrimePi[x] - RiemannR[x]]]/Log[x];
If[kk > imax, AppendTo[gg, x]; imax = kk], {x, 2, 20000}]; gg

cp's OEIS Frontend

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).

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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).

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

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A353055 Successive records of function f(x) = log(abs(pi(x) - R(x)))/log(x) where pi(x) is the number of primes <= x and R(x) is Riemann's prime counting function.

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cp's OEIS Frontend

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).

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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).

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

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Formula

A353055 Successive records of function f(x) = log(abs(pi(x) - R(x)))/log(x) where pi(x) is the number of primes <= x and R(x) is Riemann's prime counting function.

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Mathematica

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).

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).