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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, 55160980939, 155891678121, 508666658006, 1427745660374 (list; graph; refs; listen; history; text; internal format)



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


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.


Table of n, a(n) for n=1..28.

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

Anatolii A. Karatsuba and Ekatherina A. Karatsuba, The "problem of remainders" in theoretical physics: "physical zeta" function, 6th Mathematical Physics Meeting: Summer School and Conference on Modern Mathematical Physics, 14-23 September 2010, Belgrade, Serbia. [From Internet Archive Wayback Machine]

Tomás 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


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


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


from sympy import N, li, primepi, floor

def round(n):

    return int(floor(n+0.5))

def A057752(n):

    return round(N(li(10**n), 10*n)) - primepi(10**n) # Chai Wah Wu, Apr 30 2018


Cf. A006880, A052435, A057794.

Sequence in context: A018315 A146220 A054964 * A342604 A173057 A173112

Adjacent sequences:  A057749 A057750 A057751 * A057753 A057754 A057755




Robert G. Wilson v, Oct 30 2000


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

Added a(25)-a(27) using data from A006880, by Chai Wah Wu, Apr 30 2018

Term a(28) obtained using A006880. - Eduard Roure Perdices, Apr 14 2021



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