A228211 -id:A228211 - cp's OEIS frontend

A058290 Rounded difference between 10^n/(log(10^n) - A) and pi(10^n), where A is Legendre's constant and pi is the prime counting function.

-1, 4, 3, 4, 2, -4, 45, 561, 6549, 69985, 690493, 6545056, 60615397, 555560046, 5070271362, 46223804313, 421692578206, 3853431791690, 35289854434775, 323979090116197, 2981921009910364, 27516571651291205, 254562416350667928

Offset: 0

Robert G. Wilson v, Dec 07 2000

Legendre's constant is 1.08366 (A228211). - Alonso del Arte, Nov 02 2013

This sequence has historical rather than mathematical interest, cf. A228211. It is better to use 1 + 1/log(10^n) instead of A. Since A is given to only 5 decimal places, it does not make much sense to compute terms of this sequence beyond n ~ 10. For n = 9, the error a(9)/A006880(9) is about 0.14%, while the error for 1 + 1/log(10^9) instead of A is only about 0.04%. - M. F. Hasler, Dec 03 2018

Jan Gullberg, "Mathematics, From the Birth of Numbers", W. W. Norton and Company, NY and London, 1997, page 81.

Eduard Roure Perdices, Table of n, a(n) for n = 0..28 (terms 0..23 from Harry J. Smith).

Cf. A006880, A057752, A057794, A057835, A058289.

Table[ Round[ 10^n /(Log[10^n] - 1.08366) - PrimePi[10^n] ], {n, 0, 13} ]

{A006880_vec = [0, 4, 25, 168, 1229, 9592, 78498, 664579, 5761455, 50847534, 455052511 4118054813, 37607912018, 346065536839, 3204941750802, 29844570422669, 279238341033925, 2623557157654233, 24739954287740860, 234057667276344607, 2220819602560918840, 21127269486018731928, 201467286689315906290, 1925320391606803968923]} \\ Edited by M. F. Hasler, Dec 03 2018
{default(realprecision, 100); t=log(10); for (n=0, 23, write("b058290.txt", n, " ", round(10^n/(n*t - 1.08366)) - A006880_vec[n+1]))} \\ Harry J. Smith, Jun 22 2009

A058290(n)={10^n\/(n*log(10)-1.08366)-A006880(n)} \\ with A006880(n)=primepi(10^n) and/or precomputed values for n > 10. - M. F. Hasler, Dec 03 2018

a(n) = round(10^n/(log(10^n) - 1.08366)) - A006880(n). - M. F. Hasler, Dec 03 2018

More terms from Harry J. Smith, Jun 22 2009

A331020 Values k for successive maximal records of the function A defined as A(prime(k)) = log(prime(k)) - prime(k)/Pi(prime(k)) where Pi(prime(k)) is number of primes <= prime(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 15, 18, 21, 27, 28, 29, 30, 46, 61, 91, 121, 180, 184, 185, 186, 188, 189, 214, 216, 217, 257, 258, 775, 832, 1217, 1225, 1227, 1269, 1270, 1846, 1847, 2682, 2683, 2684, 2685, 2686, 2688

Offset: 1

Artur Jasinski, Jan 07 2020

This sequence is finite and complete.
Chebyshev 1852, goes on to conclude that if we put Pi(x) = x/(log(x) - A(x)) has a limit as x -> +infinity, then a limit must be 1, not 1.08366 (A228211), as Legendre incorrectly conjectured in 1808.
R. Farhadian & R. Jakimczuk 2018 prove again that the function A tends to 1 when n tends to infinity.
A(prime(2688)) = A(24137) = -24137/2688 + log(24137) = 1.11196252139...
A(n) <= -(24137/2688) + log(24137) for all positive integers n.

			   n | a(n) | A(prime(a(n)))
  ---+------+---------------
   1 |    1 | -1.306852819
   2 |    2 | -0.401387711
   3 |    3 | -0.057228754
   4 |    4 |  0.195910149
   5 |    5 |  0.197895272
   6 |    6 |  0.398282690
   7 |    7 |  0.404641915
   8 |    8 |  0.569438979
   9 |    9 |  0.579938660
  10 |   11 |  0.615805386

P. L. Chebyshev, Sur la totalité des nombres premiers inférieurs à une limite donnée, J. math. pures appl. 17, 1852 (in French).
R. Farhadian & R. Jakimczuk, One more disproof for the Legendre's conjecture regarding the prime counting function Pi(x), Notes on Number Theory and Discrete Mathematics, Vol. 24, 2018, No. 3, 84-91.
Eric Weisstein's World of Mathematics, Legendre's Constant.

Cf. A000720, A228211.

max = -2; aa = {}; Do[kk = Log[Prime[n]] - Prime[n]/PrimePi[Prime[n]];
If[kk > max, max = kk; AppendTo[aa, n]], {n, 1, 2700}]; aa

cp's OEIS Frontend

A058290 Rounded difference between 10^n/(log(10^n) - A) and pi(10^n), where A is Legendre's constant and pi is the prime counting function.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI