A059112 -id:A059112 - cp's OEIS frontend

A059111 a(n) = floor(prime(n) - n*log(n)).

2, 1, 1, 1, 2, 2, 3, 2, 3, 5, 4, 7, 7, 6, 6, 8, 10, 8, 11, 11, 9, 10, 10, 12, 16, 16, 14, 13, 11, 10, 20, 20, 21, 19, 24, 21, 23, 24, 24, 25, 26, 24, 29, 26, 25, 22, 30, 37, 36, 33, 32, 33, 30, 35, 36, 37, 38, 35, 36, 35, 32, 37, 45, 44, 41, 40, 49, 50, 54, 51, 50, 51, 53, 54, 55

Offset: 1

Henry Bottomley, Jan 04 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000
C. K. Caldwell, How Many Primes Are There?
Eric Weisstein's World of Mathematics, Prime formulas

Cf. A064658, A064659, A059112.

[ Floor(NthPrime(n)-n*Log(n)): n in [1..75] ];  // Bruno Berselli, May 02 2011

Table[Floor[Prime[n]-n Log[n]],{n,100}] (* Harvey P. Dale, May 02 2011 *)

{ default(realprecision, 100); for (n = 1, 1000, write("b059111.txt", n, " ", floor(prime(n) - n*log(n))); ) } \\ Harry J. Smith, Jun 25 2009

Definition corrected by Harry J. Smith, Jun 24 2009

A248911 a(n) = floor( prime(n) - (n+1)*log(n) ).

Original entry on oeis.org

2, 0, 0, 0, 1, 0, 1, 0, 1, 3, 2, 4, 5, 3, 3, 5, 8, 6, 8, 8, 6, 7, 7, 9, 13, 13, 10, 10, 7, 7, 17, 16, 18, 15, 21, 18, 19, 21, 20, 21, 23, 20, 25, 22, 21, 19, 26, 33, 32, 29, 28, 29, 26, 31, 32, 33, 34, 31, 32, 31, 28, 32, 41, 40, 37, 36, 45, 45, 50, 47, 46, 46

Offset: 1

Michel Lagneau, Oct 16 2014

nonn

The function log gives the natural logarithm (to base e).

See A059111 for the sequence a(n) = floor(prime(n)-n*log(n)).

			a(8) = 0 because floor(prime(8)-(8+1)*log(8)) = floor(19 -9*2.07944154...) = floor(.28502612...) = 0.

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A000040, A059111, A059112.

[Floor(NthPrime(n)-(n+1)*Log(n)): n in [1..80]]; // Vincenzo Librandi, Oct 16 2014

with(numtheory):
for n from 1 to 200 do:
  p:=floor(evalf(ithprime(n)-(n+1)*ln(n))): printf(`%d, `,p):
od:

Table[Floor[Prime[n]-(n+1)Log[n]], {n, 100}]

A204594 Nearest integer to nlog(n) + nlog(log(n)) - n + n/log(n)(log(log(n))-2) - nlog(log(n))*(log(log(n))-6)/(2 log(n)^2), an asymptotic expression for prime(n).

Original entry on oeis.org

-13, -4, 0, 3, 6, 10, 13, 17, 20, 24, 28, 32, 36, 40, 44, 49, 53, 57, 62, 66, 71, 76, 80, 85, 90, 95, 100, 105, 109, 115, 120, 125, 130, 135, 140, 145, 151, 156, 161, 167, 172, 177, 183, 188, 194, 199, 205, 210, 216, 222, 227, 233, 239, 244, 250, 256, 262

Offset: 2

M. F. Hasler, Feb 07 2012

sign

Pierre Dusart, Estimates of Some Functions Over Primes without R.H. (2010).
Wikipedia, Prime number theorem

Cf. A059112, A059111, A064658, A064659.

Table[Round[n Log[n] + n Log[Log[n]] - n + n/Log[n](Log[Log[n]] - 2) - n Log[Log[n]](Log[Log[n]] - 6)/(2 Log[n]^2)], {n, 2, 58}] (* Alonso del Arte, Feb 07 2012 *)

A204594(n)=round(n*log(n)+n*log(log(n))-n+n/log(n)*(log(log(n))-2)-n*log(log(n))/2/log(n)^2*(log(log(n))-6))

A248912 a(n) = floor(prime(n) - (n+1)*(log(n) + log(log(n))) + n) for n > 1.

Original entry on oeis.org

4, 3, 2, 3, 2, 3, 1, 2, 4, 2, 4, 4, 2, 2, 4, 6, 3, 5, 5, 2, 3, 3, 4, 7, 7, 4, 3, 0, -1, 8, 7, 8, 5, 10, 7, 7, 8, 7, 8, 8, 5, 10, 6, 5, 1, 8, 14, 13, 9, 8, 8, 5, 9, 9, 10, 10, 6, 7, 5, 1, 5, 13, 12, 8, 6, 14, 14, 18, 14, 12, 12, 14, 14, 14, 12, 12, 14, 12, 14

Offset: 2

Michel Lagneau, Oct 16 2014

sign

The function log gives the natural logarithm (to base e).
a(30) = -1 is the unique negative value.
See A059112 for floor( prime(n) - n*log(n) - n*log(log(n)) + n ) where prime(n) is the n-th prime.

			a(8) = 1 because floor(prime(8)-(8+1)*(log(8)+log(log(8))) + 8) = floor(19-9*(2.0794415...+0.7320993...) + 8) = floor(1.6961318...) = 1.

Michel Lagneau, Table of n, a(n) for n = 2..20000

Cf. A000040, A059111, A059112, A248911.

[Floor(NthPrime(n)-(n+1)*(Log(n)+Log(Log(n)))+n): n in [2..80]]; // Vincenzo Librandi, Oct 16 2014

with(numtheory):for n from 1 to 200 do:p:=floor(evalf(ithprime(n)-(n+1)*ln(n)- (n+1)*ln(ln(n)) + n)): printf(`%d, `,p):od:

Table[Floor[Prime[n]-(n+1)*(Log[n]+Log[Log[n]])+n], {n,2, 100}]

a(n) = floor(prime(n)-(n+1)*(log(n)+log(log(n))) + n); \\ Michel Marcus, Mar 05 2022

cp's OEIS Frontend

A059111 a(n) = floor(prime(n) - n*log(n)).

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A248911 a(n) = floor( prime(n) - (n+1)*log(n) ).

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A204594 Nearest integer to n*log(n) + n*log(log(n)) - n + n/log(n)*(log(log(n))-2) - n*log(log(n))*(log(log(n))-6)/(2 log(n)^2), an asymptotic expression for prime(n).

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A248912 a(n) = floor(prime(n) - (n+1)*(log(n) + log(log(n))) + n) for n > 1.

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Author

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Magma

Maple

Mathematica

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A204594 Nearest integer to nlog(n) + nlog(log(n)) - n + n/log(n)(log(log(n))-2) - nlog(log(n))*(log(log(n))-6)/(2 log(n)^2), an asymptotic expression for prime(n).