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%I #41 Feb 02 2024 16:14:12
%S 1,4,9,16,37,129,338,753,1700,3103,11587,38262,108970,314889,1052618,
%T 3214631,7956588,21949554,99877774,222744643,597394253,1932355207,
%U 7250186215,17146907277,55160980938,155891678120,508666658005,1427745660373,4551193622463
%N Differences between the prime-counting function and Gauss's approximation for number of primes < 10^n.
%C From _Vladimir Pletser_, Mar 16 2013: (Start)
%C As Li(2) = 1.04516..., a(n) = A057752(n) - 1.
%C 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.
%C 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.
%C 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):
%C - 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;
%C - E. Weinstein in "Prime Counting Function" gives also values of -(A057752) for pi(10^n)-Li(10^n)
%C - Wikipedia gives a Table with Li(10^n)-pi(10^n) (A057752);
%C - 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)
%D Jonathan Borwein, David H. Bailey, Mathematics by Experiment, A. K. Peters, 2004, p. 65 (Table 2.2).
%D John H. Conway and R. K. Guy, The Book of Numbers, Copernicus, an imprint of Springer-Verlag, NY, 1996, page 144.
%H C. K. Caldwell, <a href="https://t5k.org/howmany.shtml">How Many Primes Are There?</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeCountingFunction.html">Prime Counting Function</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LogarithmicIntegral.html">Logarithmic Integral</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Prime-counting_function">Prime counting function</a>
%F 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.
%F a(n) = A190802(n) - A006880(n).
%e Given x = 10^4, pi(x) = 1229, Gauss's approximation = 1245. Thus a(4) = 1245 - 1229 = 16.
%t Table[Round[Integrate[1/Log[t],{t,2,10^n}]]-PrimePi[10^n],{n,27}] (* _James C. McMahon_, Feb 01 2024 *)
%Y Cf. A057754, A057752, A006880, A190802, A106313, A223166, A223167.
%K nonn
%O 1,2
%A _Gary W. Adamson_, Apr 28 2005
%E a(23)-a(24) from _Nathaniel Johnston_, May 25 2011
%E a(25)-a(28), obtained using A006880, added by _Eduard Roure Perdices_, Apr 16 2021
%E a(29) (using A006880) from _Alois P. Heinz_, Feb 01 2024
%E Name clarified by _James C. McMahon_, Feb 02 2024
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