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A057752 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). 10
2, 5, 10, 17, 38, 130, 339, 754, 1701, 3104, 11588, 38263, 108971, 314890, 1052619, 3214632, 7956589, 21949555, 99877775, 222744644, 597394254, 1932355208, 7250186216, 17146907278 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

REFERENCES

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.

LINKS

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

MATHEMATICA

Table[Round[LogIntegral[10^n] - PrimePi[10^n]], {n, 1, 13}]

PROG

(PARI) A057752=vector(#A006880, i, round(-eint1(-log(10^i))-A006880[i])) - M. F. Hasler, Feb 26 2008

CROSSREFS

Cf. A006880, A052435, A057794.

Sequence in context: A018315 A146220 A054964 * A173057 A213299 A172114

Adjacent sequences:  A057749 A057750 A057751 * A057753 A057754 A057755

KEYWORD

sign,hard

AUTHOR

Robert G. Wilson v, Oct 30 2000

EXTENSIONS

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

STATUS

approved

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Last modified November 28 14:16 EST 2014. Contains 250361 sequences.