A106313 -id:A106313 - cp's OEIS frontend

A190802 Gauss' approximation for the number of primes below 10^n.

5, 29, 177, 1245, 9629, 78627, 664917, 5762208, 50849234, 455055614, 4118066400, 37607950280, 346065645809, 3204942065691, 29844571475287, 279238344248556, 2623557165610821, 24739954309690414, 234057667376222381, 2220819602783663483

Offset: 1

Nathaniel Johnston, May 25 2011

nonn

The offset logarithmic integral or Eulerian logarithmic integral Li(10^n)-Li(2), i.e., integral(2..x, dt/log(t)), appears in Gauss’s formula for counting prime numbers < 10^n and is sometimes referred to as the "European" definition. - Vladimir Pletser, Mar 17 2013

Jonathan Borwein, David H. Bailey, "Mathematics by Experiment", A. K. Peters, 2004, p. 65 (Table 2.2).

Vladimir Pletser, Table of n, a(n) for n = 1..500
Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, and Stephan Ramon Garcia, The Bateman-Horn Conjecture: Heuristics, History, and Applications, arXiv:1807.08899 [math.NT], 2018-2019. See Table 1 p. 6.

Cf. A006880, A106313.

seq(round(evalf(integrate(1/log(t),t=2..10^n))), n=1..21);

Table[Round[Integrate[1/Log[t],{t,2,10^n}]],{n,20}] (* James C. McMahon, Feb 06 2024 *)

a(n) = round(integral(dt/log(t),t=2..10^n)).

A223167 Difference between nearest integer to (Li(10^n)-Li(3)) and pi(10^n), where Li(10^n)-Li(3) = integral(3.. 10^n, dt/log(t)) (A223166) and pi(10^n) = number of primes <= 10^n (A006880).

Original entry on oeis.org

0, 3, 7, 15, 36, 127, 337, 752, 1699, 3101, 11585, 38261, 108969, 314888, 1052616, 3214630, 7956587, 21949553, 99877773, 222744641, 597394252, 1932355206, 7250186214, 17146907276, 55160980937, 155891678119, 508666658004, 1427745660372

Offset: 1

Vladimir Pletser, Mar 16 2013

nonn

As Li(3)= 2.163588..., A057752(n)-a(n) = 2, except for n =3, 6, 10, 11, 15, 20 where A057752(n)-a(n)= 3.

This sequence yields an even better average relative difference than Gauss's approximation (A106313), i.e., Average(a(n)/pi(10^n)) = 7.4969...*10^-3 for 1<=n<=24, compared to Average(A057752(n)/pi(10^n)) = 3.2486...*10^-2 and Average(A106313(n)/pi(10^n)) = 2.0116...*10^-2, showing that, when using the logarithmic integral, Li(10^n)-Li(3) (A223166) gives a better approximation to pi(10^n) than Li(10^n)-Li(2) (A190802) and than Li(10^n) (A057754).

Eric Weisstein's World of Mathematics, Prime Counting Function
Eric Weisstein's World of Mathematics, Logarithmic Integral

Cf. A057754, A057752, A006880, A190802, A106313.

a[n_] := Round[LogIntegral[10^n] - LogIntegral[3]] - PrimePi[10^n]; Table[a[n], {n, 1, 14}]

a(n)=round(eint1(-log(3))-eint1(-n*log(10)))-primepi(10^n) \\ Charles R Greathouse IV, May 03 2013

a(n) = A223166(n) - A006880(n).

Terms a(25)-a(28) obtained using A006880. - Eduard Roure Perdices, Apr 14 2021

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).

Original entry on oeis.org

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).

cp's OEIS Frontend

A190802 Gauss' approximation for the number of primes below 10^n.

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A223167 Difference between nearest integer to (Li(10^n)-Li(3)) and pi(10^n), where Li(10^n)-Li(3) = integral(3.. 10^n, dt/log(t)) (A223166) and pi(10^n) = number of primes <= 10^n (A006880).

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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).

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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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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).