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A057752
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Difference between nearest integer to Li(10^n) and pi(10^n), where Li(x) = integral of log(x) and pi(10^n) = number of primes <= 10^n (A006880).
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8
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2, 5, 10, 17, 38, 130, 339, 754, 1701, 3104, 11588, 38263, 108971, 314890, 1052619, 3214632, 7956589, 21949555, 99877775, 222744644, 597394254, 1932355208, 7250186216, 17146907278
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OFFSET
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1,1
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COMMENTS
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On his prime pages C. K. Caldwell remarks: "However in 1914 Littlewood proved that pi(x)-Li(x) assumes both positive and negative values infinitely often". - Frank Ellermann, May 31, 2003
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REFERENCES
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John H. Conway and R. K. Guy, The Book of Numbers, Copernicus, an imprint of Springer-Verlag, NY, 1995, page 146.
Marcus du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see table on p. 90.
Anatolii A. Karatsuba and Ekatherina A. Karatsuba, The "problem of remainders" in theoretical physics: "physical zeta" function, http://www.phy.bg.ac.rs/~mphys6/proceedings5/14-KaratsubaKaratsuba.ps.
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LINKS
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Table of n, a(n) for n=1..24.
Chris K. Caldwell, How many primes are there, table, Values of pi(x).
Chris K. Caldwell, How many primes are there, table, Approximations to pi(x).
Xavier Gourdon & Pascal Sebah, Counting the primes
Andrew Granville, Harald Cramer and the Distribution of Prime Numbers
Tomas Oliveira e Silva, Tables of values of pi(x) and of pi2(x)
Y. Saouter, P. Demichel, A sharp region where pi(x)-li(x) is positive, Math. Comp. 79 (272) (2010) 2395-2405. [From R. J. Mathar, Oct 08 2010]
Munibah Tahir, A new bound for the smallest x with pi(x) > li(x) (2010)
Eric Weisstein's World of Mathematics, Prime Counting Function
Wikipedia, Prime number theorem
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MATHEMATICA
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Table[Round[LogIntegral[10^n] - PrimePi[10^n]], {n, 1, 13}]
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PROG
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(PARI) A057752=vector(#A006880, i, round(-eint1(-log(10^i))-A006880[i])) - M. F. Hasler, Feb 26 2008
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CROSSREFS
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Cf. A006880, A052435, A045916, A057794.
Sequence in context: A018315 A146220 A054964 * A173057 A213299 A172114
Adjacent sequences: A057749 A057750 A057751 * A057753 A057754 A057755
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KEYWORD
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sign,hard
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AUTHOR
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Robert G. Wilson v, Oct 30 2000
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EXTENSIONS
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More terms from Frank Ellermann, May 31, 2003
The value of a(23) is not known at present, I believe. - N. J. A. Sloane, Mar 17 2008
Name corrected and extended for last two terms a(23) and a(24), with Pi(10^n) for n=23 and 24 from A006880, by Vladimir Pletser, Mar 10 2013
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STATUS
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approved
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