A190802 -id:A190802 - cp's OEIS frontend

A228067 Difference of consecutive integers nearest to Li(10^n) - Li(2), where Li(x) = integral(0..x, dt/log(t)) (A190802, known as Gauss' approximation for the number of primes below 10^n).

5, 24, 148, 1068, 8384, 68998, 586290, 5097291, 45087026, 404206380, 3663010786, 33489883880, 308457695529, 2858876419882, 26639629409596, 249393772773269, 2344318821362265, 22116397144079593, 209317713066531967, 1986761935407441102

Offset: 1

Vladimir Pletser, Aug 06 2013

nonn

This sequence gives a good approximation of the number of primes with n digits (A006879); see (A228068).

Note that A190802(n)=(Li(10^n)-Li(2)) is not defined for n=0. Its value is arbitrarily set to 0.

			For n = 1, A190802(1) - A190802(0) = 5-0 = 5.

Vladimir Pletser, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Prime Counting Function
Eric Weisstein's World of Mathematics, Logarithmic Integral

Cf. A006879, A190802, A228068, A228065.

a(n) = A190802(n) - A190802(n-1).

A106313 Differences between the prime-counting function and Gauss's approximation for number of primes < 10^n.

Original entry on oeis.org

1, 4, 9, 16, 37, 129, 338, 753, 1700, 3103, 11587, 38262, 108970, 314889, 1052618, 3214631, 7956588, 21949554, 99877774, 222744643, 597394253, 1932355207, 7250186215, 17146907277, 55160980938, 155891678120, 508666658005, 1427745660373, 4551193622463

Offset: 1

Gary W. Adamson, Apr 28 2005

nonn

From Vladimir Pletser, Mar 16 2013: (Start)
As Li(2) = 1.04516..., a(n) = A057752(n) - 1.
This sequence gives the exact values of the difference between Gauss's Li (defined as integral(2..10^n, dt/log(t)) or Li(10^n)-Li(2)) and the number of primes <= 10^n (A006880). For large values of x=10^n, Li(2) can be neglected but for small values of x=10^n, the value of Li(2) cannot be neglected.
This sequence yields a better average relative difference, i.e., average(a(n)/pi(10^n)) = 2.0116...x10^-2 for 1<=n<=24, compared to average(A057752(n)/pi(10^n)) = 3.2486...x10^-2. However see also Li(10^n)-Li(3) in A223166 and A223167.
Note that most of the Tables in the literature giving the difference of Li(10^n) - pi(10^n) use the values of A057752 as the difference between Gauss's Li values and pi(10^n). This is incorrect and the values above should be used instead. For example (certainly not exhaustive):
- John H. Conway and R. K. Guy in "The Book of Numbers" show in Fig. 5.2, p. 144, Li(N) as integral(2..10^n, dt/log(t)) but reports values of A057752 (the difference of integral(0..10^n, dt/log(t)) and pi(10^n)) in Table 5.2, p. 146;
- Eric Weisstein in "Prime Counting Function" gives also values of -(A057752) for pi(10^n)-Li(10^n)
- Wikipedia gives a Table with Li(10^n)-pi(10^n) (A057752);
- C. K. Caldwell in Table 3 in the link below give values of Li(10^n) while values of Li(10^n) - Li(2) would be more suited. (End)

			Given x = 10^4, pi(x) = 1229, Gauss's approximation = 1245. Thus a(4) = 1245 - 1229 = 16.

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.

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. A057754, A057752, A006880, A190802, A106313, A223166, A223167.

Table[Round[Integrate[1/Log[t],{t,2,10^n}]]-PrimePi[10^n],{n,27}] (* James C. McMahon, Feb 01 2024 *)

The prime counting function pi(x) runs through x = 10^1, 10^2, 10^3, ...; being subtracted from Gauss's approximation, integral(2, x)dt/log t.

a(n) = A190802(n) - A006880(n).

a(23)-a(24) from Nathaniel Johnston, May 25 2011
a(25)-a(28), obtained using A006880, added by Eduard Roure Perdices, Apr 16 2021
a(29) (using A006880) from Alois P. Heinz, Feb 01 2024
Name clarified by James C. McMahon, Feb 02 2024

A223166 Integer nearest to Li(10^n) - Li(3), where Li(x) = integral(0..x, dt/log(t)).

Original entry on oeis.org

4, 28, 175, 1244, 9628, 78625, 664916, 5762207, 50849233, 455055612, 4118066398, 37607950279, 346065645808, 3204942065690, 29844571475285, 279238344248555, 2623557165610820, 24739954309690413, 234057667376222380, 2220819602783663481

Offset: 1

Vladimir Pletser, Mar 16 2013

nonn

Vladimir Pletser, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Prime Counting Function
Eric Weisstein's World of Mathematics, Logarithmic Integral

Cf. A057754, A006880, A190802.

seq(round(evalf(integrate(1/log(t), t=3..10^n))), n=1..1000);

Table[Round[LogIntegral[10^n]-LogIntegral[3]],{n,30}] (* Harvey P. Dale, Aug 24 2022 *)

a(n) = round(Li( 10^n)-Li(3)).

A228068 Difference between the number of primes with n digits (A006879) and the difference of consecutive integers nearest to Li(10^n) - Li(2) (see A228067).

Original entry on oeis.org

-1, -3, -5, -7, -21, -92, -209, -415, -947, -1403, -8484, -26675, -70708, -205919, -737729, -2162013, -4741957, -13992966, -77928220, -122866869, -374649610, -1334960954, -5317831008, -9896721062, -38014073661

Offset: 1

Vladimir Pletser, Aug 06 2013

sign

The sequence A006879(n) is always < A228067(n) for 1 <= n <= 25.
The sequence (A228067) yields an average relative difference in absolute value, i.e., average(abs(A228068(n))/A006879(n) = 0.0175492... for 1 <= n <= 25.
Note that A190802(n) = (Li(10^n) - Li(2)) is not defined for n=0. Its value is set arbitrarily to 0.

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

Cf. A006880, A006879, A228067, A228066.

a(n) = A006879(n) - A228067(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).

A226945 Integer nearest f(10^n), where f(x) = Sum of ( mu(k) * H(k)/k^(3/2) * Integral Log(x^(1/k)) ) for k = 1 to infinity, where H(k) is the harmonic number sum_{i=1..k} 1/i.

Original entry on oeis.org

4, 25, 168, 1226, 9585, 78521, 664652, 5761512, 50847348, 455050385, 4118051652, 37607908133, 346065524108, 3204941711340, 29844570436484, 279238341185832, 2623557156537070, 24739954282695698, 234057667295619287, 2220819602542218793

Offset: 1

Arkadiusz Wesolowski, Aug 31 2013

nonn

The sequence gives exactly the values of pi(10^n) for n = 1 to 3.

A228724 gives the difference between A006880 and this sequence.

David Baugh, Table of n, a(n) for n = 1..100
Chris K. Caldwell, How Many Primes Are There?
Eric Weisstein's World of Mathematics, Prime Counting Function
Eric Weisstein's World of Mathematics, Prime Number Theorem

Cf. A006880, A057834, A226744, A057754, A190802, A057793, A201542, A228724.

f[n_Integer] := Sum[N[MoebiusMu[k]*HarmonicNumber[k]/k^(3/2)*LogIntegral[n^(1/k)], 50], {k, 5!}]; Table[Round[f[10^n]], {n, 20}]

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

A228067 Difference of consecutive integers nearest to Li(10^n) - Li(2), where Li(x) = integral(0..x, dt/log(t)) (A190802, known as Gauss' approximation for the number of primes below 10^n).

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A106313 Differences between the prime-counting function and Gauss's approximation for number of primes < 10^n.

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A223166 Integer nearest to Li(10^n) - Li(3), where Li(x) = integral(0..x, dt/log(t)).

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A228068 Difference between the number of primes with n digits (A006879) and the difference of consecutive integers nearest to Li(10^n) - Li(2) (see A228067).

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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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A226945 Integer nearest f(10^n), where f(x) = Sum of ( mu(k) * H(k)/k^(3/2) * Integral Log(x^(1/k)) ) for k = 1 to infinity, where H(k) is the harmonic number sum_{i=1..k} 1/i.

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Mathematica

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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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).