A057834 -id:A057834 - cp's OEIS frontend

A228065 Difference of consecutive integers nearest to (10^n)/log(10^n) (A057834).

4, 18, 123, 941, 7600, 63696, 548039, 4808260, 42826261, 386039540, 3513837172, 32243075171, 297881471562, 2768030763779, 25850862018051, 242481085729315, 2283239371770773, 21572797793887019, 204448571890127322, 1942896366409284492

Offset: 1

Vladimir Pletser, Aug 06 2013

nonn

This sequence gives an approximation of the number of primes with n digits (A006879); see A228066.

Note that A057834(n) = (10^n)/log(10^n) is not defined for n=0. Its value is set arbitrarily to 0.

			For n = 1, A057834(1) - A057834(0) = 4-0 = 4.

Vladimir Pletser, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Prime Counting Function.

Cf. A006879, A057834, A228066.

a(n) = A057834(n) - A057834(n-1).

A057835 Difference between pi(10^n) and the integer nearest to 10^n / log(10^n).

Original entry on oeis.org

0, 3, 23, 143, 906, 6116, 44158, 332774, 2592592, 20758029, 169923159, 1416705193, 11992858452, 102838308636, 891604962452, 7804289844393, 68883734693928, 612483070893536, 5481624169369961, 49347193044659702, 446579871578168707, 4060704006019620994, 37083513766578631309, 339996354713708049069, 3128516637843038351228

Offset: 1

Robert G. Wilson v, Nov 08 2000

nonn

John H. Conway and R. K. Guy, "The Book of Numbers," Copernicus, an imprint of Springer-Verlag, NY, 1995, Page 146.

Eduard Roure Perdices, Table of n, a(n) for n = 1..28
Wikipedia, Prime number theorem

Table[ PrimePi[10^n] - Round[ N[ 10^n/Log[ 10^n ] ] ], {n, 1, 13} ]

a(n)=primepi(10^n)-round(10^n/log(10^n)) \\ Charles R Greathouse IV, Mar 22 2015

a(n) = A006880(n) - A057834(n). - Henry Bottomley, Aug 10 2005

a(n) ~ 10^n/(n log 10)^2. - Charles R Greathouse IV, Mar 22 2015

More terms from Jud McCranie, Jun 21 2005
Corrected and extended by Henry Bottomley, Aug 10 2005
a(22) to a(25) from Vladimir Pletser, Aug 10 2013

A228066 a(n) = A006879(n) - A228065(n).

Original entry on oeis.org

0, 3, 20, 120, 763, 5210, 38042, 288616, 2259818, 18165437, 149165130, 1246782034, 10576153259, 90845450184, 788766653816, 6912684881941, 61079444849535, 543599336199608, 4869141098476425, 43865568875289741, 397232678533509005, 3614124134441452287

Offset: 1

Vladimir Pletser, Aug 06 2013

Difference between the number of primes with n digits (A006879) and the difference of consecutive integers nearest to (10^n)/log(10^n) (see A228065).
The sequence A006879(n) is always > A228065(n) for 1 <= n <= 28.
The sequence (A228065) provides exactly the first value of pi(10^n)- pi(10^(n-1)) for n = 1, and yields an average relative difference in absolute value, i.e., average(abs(A228066(n))/(A006879(n))) = 0.0436296... for 1 <= n <= 28.
Note that A057834(n) = 10^n/log(10^n) is not defined for n = 0; its value is set arbitrarily to 0. - Updated by Eduard Roure Perdices, Apr 18 2021

Eduard Roure Perdices, Table of n, a(n) for n = 1..28
Eric Weisstein's World of Mathematics, Prime Counting Function

Cf. A006880, A006879, A228065.

a(n) = A006879(n) - A228065(n).

A058289 Integer nearest 10^n/(log(10^n) - 1.08366).

Original entry on oeis.org

-1, 8, 28, 172, 1231, 9588, 78543, 665140, 5768004, 50917519, 455743004, 4124599869, 37668527415, 346621096885, 3210012022164, 29890794226982, 279660033612131, 2627410589445923, 24775244142175635, 234381646366460804

Offset: 0

Robert G. Wilson v, Dec 07 2000

sign

"Adrien-Marie Legendre in 1778 published his work 'Essai sur la théorie des nombres' where he proposed a modified form of the first approximation, pi(n) ~ n/ln n." (Gullberg)

Jan Gullberg, "Mathematics, From the Birth of Numbers," W. W. Norton and Company, NY and London, 1997, page 80.

Harry J. Smith, Table of n, a(n) for n = 0..500

Cf. A006880, A057754, A057793, A057834, A058290.

Table[ Round[ 10^n /(Log[10^n] - 1.08366) ], {n, 0, 22} ]

{ default(realprecision, 1000); t=log(10); for (n=0, 500, write("b058289.txt", n, " ", round(10^n/(n*t - 1.08366))); ); } \\ Harry J. Smith, Jun 22 2009

Corrected some terms. - 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)).

A226945 Integer nearest f(10^n), where f(x) = Sum of ( mu(k) * H(k)/k^(3/2) * Integral Log(x^(1/k)) ) for k = 1 to infinity, where H(k) is the harmonic number sum_{i=1..k} 1/i.

Original entry on oeis.org

4, 25, 168, 1226, 9585, 78521, 664652, 5761512, 50847348, 455050385, 4118051652, 37607908133, 346065524108, 3204941711340, 29844570436484, 279238341185832, 2623557156537070, 24739954282695698, 234057667295619287, 2220819602542218793

Offset: 1

Arkadiusz Wesolowski, Aug 31 2013

nonn

The sequence gives exactly the values of pi(10^n) for n = 1 to 3.

A228724 gives the difference between A006880 and this sequence.

David Baugh, Table of n, a(n) for n = 1..100
Chris K. Caldwell, How Many Primes Are There?
Eric Weisstein's World of Mathematics, Prime Counting Function
Eric Weisstein's World of Mathematics, Prime Number Theorem

Cf. A006880, A057834, A226744, A057754, A190802, A057793, A201542, A228724.

f[n_Integer] := Sum[N[MoebiusMu[k]*HarmonicNumber[k]/k^(3/2)*LogIntegral[n^(1/k)], 50], {k, 5!}]; Table[Round[f[10^n]], {n, 20}]

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}]

cp's OEIS Frontend

A228065 Difference of consecutive integers nearest to (10^n)/log(10^n) (A057834).

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A057835 Difference between pi(10^n) and the integer nearest to 10^n / log(10^n).

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A228066 a(n) = A006879(n) - A228065(n).

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A058289 Integer nearest 10^n/(log(10^n) - 1.08366).

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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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A226945 Integer nearest f(10^n), where f(x) = Sum of ( mu(k) * H(k)/k^(3/2) * Integral Log(x^(1/k)) ) for k = 1 to infinity, where H(k) is the harmonic number sum_{i=1..k} 1/i.

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

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