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A106313
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Differences between the prime-counting function and Gauss's approximation for number of primes < 10^n.
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6
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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
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OFFSET
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1,2
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COMMENTS
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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)
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REFERENCES
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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Given x = 10^4, pi(x) = 1229, Gauss's approximation = 1245. Thus a(4) = 1245 - 1229 = 16.
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MATHEMATICA
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Table[Round[Integrate[1/Log[t], {t, 2, 10^n}]]-PrimePi[10^n], {n, 27}] (* James C. McMahon, Feb 01 2024 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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