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

0,2

Comments

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

References

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

Links

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

Formula

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

Mathematica

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

Prog

(PARI) {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
(PARI) 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

Crossrefs

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

Sequence in context: A161882 A276789 A082125 * A356033 A002285 A106049

Adjacent sequences: A058287 A058288 A058289 * A058291 A058292 A058293

Keyword

sign,less

Author

Robert G. Wilson v, Dec 07 2000

Extensions

More terms from Harry J. Smith, Jun 22 2009

Status

approved