A058289 -id:A058289 - 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

A193257 Floor((10^n)/(log(10^n) - 1)).

Original entry on oeis.org

7, 27, 169, 1217, 9512, 78030, 661458, 5740303, 50701542, 454011971, 4110416300, 37550193649, 345618860220, 3201414635780, 29816233849000, 279007258230819, 2621647966812031, 24723998785919976, 233922961602470390, 2219671974013732243

Offset: 1

Arkadiusz Wesolowski, Jul 19 2011

nonn

lim n -> infinity (log(n) - n/pi(n)) = 1, where pi(n) is the prime counting function.

			a(2) = 27 because (10^2)/(log(10^2) - 1) = 27.7379415786....

A. M. Legendre, Essai sur la Théorie des Nombres, Paris: Duprat, 1808.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Legendre's Constant
Eric Weisstein's World of Mathematics, Prime Counting Function
Eric Weisstein's World of Mathematics, Prime Number Theorem

Another version of A226744.

Cf. A058289, A006880, A057834, A000720.

[Floor(10^n/(Log(10^n)-1)) : n in [1..20]]

Table[Floor[10^n/(Log[10^n] - 1)], {n, 20}]

for(n=1, 20, print1(floor(10^n/(log(10^n)-1)), ", "))

a(n)=10^n\(n*log(10)-1) \\ Charles R Greathouse IV, Jul 30 2011

a(n) = floor((10^n)/(log(10^n) - 1)).

A226744 Round((10^n)/(log(10^n) - 1)).

Original entry on oeis.org

8, 28, 169, 1218, 9512, 78030, 661459, 5740304, 50701542, 454011971, 4110416301, 37550193650, 345618860221, 3201414635781, 29816233849001, 279007258230820, 2621647966812031, 24723998785919976, 233922961602470391, 2219671974013732243

Offset: 1

Arkadiusz Wesolowski, Aug 31 2013

nonn

			a(2) = 28 because (10^2)/(log(10^2) - 1) = 27.7379415786....

A. M. Legendre, Essai sur la Théorie des Nombres, Paris: Duprat, 1808.

Eric Weisstein's World of Mathematics, Legendre's Constant
Eric Weisstein's World of Mathematics, Prime Counting Function
Eric Weisstein's World of Mathematics, Prime Number Theorem

Another version of A193257.

Cf. A058289, A006880, A057834, A000720.

Table[Round[10^n/(Log[10^n] - 1)], {n, 20}]

for(n=1, 20, print1(round(10^n/(log(10^n)-1)), ", "));

a(n) = round((10^n)/(log(10^n) - 1)).

A135325 Smallest n such that pi(n)=Floor[n/log((n-pi(n))/e)].

Original entry on oeis.org

59473, 59671, 59699, 59707, 59729, 59743, 59747, 59755, 59771, 59779, 59791, 59799, 59810, 152993, 153001, 155801, 155809, 155821, 155833, 155849, 155853, 155877, 155889, 155913, 155925, 210533, 211891, 211933, 211943, 211949, 211969

Offset: 1

Manuel Valdivia, Dec 07 2007

nonn

Eric Weisstein's World of Mathematics, Prime Number Theorem.

Cf. A057834, A058289, A000720.

j=0;Do[k=Floor[n/Log[(n-PrimePi[n])/E]];If[PrimePi[n]==k&&k>j,Print[n];j=k],{n,1,10^7,1}]

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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.

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A193257 Floor((10^n)/(log(10^n) - 1)).

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A226744 Round((10^n)/(log(10^n) - 1)).

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A135325 Smallest n such that pi(n)=Floor[n/log((n-pi(n))/e)].

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