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

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

1,2

Comments

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;

- E. Weinstein 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)

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..29.

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

Formula

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).

Example

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

Mathematica

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

Crossrefs

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

Sequence in context: A231180 A250029 A111378 * A219355 A291216 A074101

Adjacent sequences: A106310 A106311 A106312 * A106314 A106315 A106316

Keyword

nonn

Author

Gary W. Adamson, Apr 28 2005

Extensions

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

Status

approved