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

Author

Gary W. Adamson, Apr 28 2005

Keywords

• nonn

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;

- 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)

Examples

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

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

• 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

Crossrefs

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

Programs

Mathematica

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

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).

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