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).

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, 4551193622464

Offset: 1

Author

Robert G. Wilson v, Oct 30 2000

Keywords

• sign
• hard

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.

Links

• 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 and 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 and 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

Crossrefs

Cf. A006880, A052435, A057794.

Programs

Mathematica

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

Python

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

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

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

a(28) added, using data from A006880, by Eduard Roure Perdices, Apr 14 2021

a(29) added, using data from A006880, by Reza K Ghazi, May 10 2022