A064658 -id:A064658 - 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

A064659 a(n) = round(prime(n) - n*log(n)).

Original entry on oeis.org

2, 2, 2, 1, 3, 2, 3, 2, 3, 6, 5, 7, 8, 6, 6, 9, 11, 9, 11, 11, 9, 11, 11, 13, 17, 16, 14, 14, 11, 11, 21, 20, 22, 19, 25, 22, 23, 25, 24, 25, 27, 24, 29, 26, 26, 23, 30, 37, 36, 33, 32, 34, 31, 36, 37, 38, 39, 35, 36, 35, 32, 37, 46, 45, 42, 40, 49, 50, 55

Offset: 1

N. J. A. Sloane, Oct 10 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A064658, A059111.

{ default(realprecision, 100); for (n = 1, 1000, write("b064659.txt", n, " ", round(prime(n) - n*log(n))); ) } \\ Harry J. Smith, Jun 25 2009

a(n) ~ n*log(log(n)) * (1 + 1/log(n) - 1/log(log(n)) - 2/(log(n)*log(log(n)))). - Vaclav Kotesovec, May 23 2020

A272231 a(n) = round(n / pi(n)), where pi(n) is the prime-counting function.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 3, 4, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 2

Benjamin Przybocki, Apr 22 2016

nonn

This sequence grows slowly; a(n) reaches 10 at n = 39017.

Cf. A000720, A064658, A107609.

Table[Round[n/PrimePi@ n], {n, 2, 120}] (* Michael De Vlieger, May 20 2016 *)

n \/ primepi(n) \\ Charles R Greathouse IV, Apr 30 2016

a(n) = round(n / pi(n)).

a(n) ~ log(n). - Charles R Greathouse IV, Apr 30 2016

A204325 Residual of an asymptotic formula for the n-th prime: a(n) = floor(prime(n)-nlog(n) + n - nlog(log(n)) - (n/log(n))(log(log(n)) - 2) + (log(log(n)) - 6)nlog(log(n))/(2log(n)^2)).

Original entry on oeis.org

16, 8, 7, 7, 6, 7, 5, 6, 8, 6, 9, 9, 7, 6, 8, 10, 8, 9, 9, 6, 8, 7, 8, 11, 11, 8, 7, 4, 3, 12, 11, 12, 9, 14, 10, 11, 12, 11, 11, 12, 9, 13, 10, 8, 5, 11, 18, 16, 12, 11, 11, 8, 12, 12, 12, 13, 9, 9, 7, 4, 8, 16, 14, 10, 8, 16, 16, 20, 16

Offset: 2

José María Grau Ribas, Jan 14 2012

sign

prime(n) ~ n*log(n) + n - n*log(log(n)) - (n/log(n))*(log(log(n)) - 2) + (log(log(n)) - 6)*n*log(log(n))/(2*log(n)^2).

The first negative term is a(214) = -2. - Jason Yuen, Feb 17 2025

M. Cipolla, La determinazione asintotica dell'n-mo numero primo, Rend. d. R. Acc. di sc. fis. e mat. di Napoli, s. 3, VIII (1902), pp. 132-166.

Michel Marcus, Table of n, a(n) for n = 2..10000
Pierre Dusart, Estimates of Some Functions Over Primes without R.H., arXiv:1002.0442 [math.NT], 2010.
Wikipedia, Prime number theorem

Cf. A064658, A059111.

Table[Floor[Prime[n]-n*Log[n]+n-n*Log[Log[n]]- (n/Log[n]) (Log[Log[n]]-2)+(Log[Log[n]]-6)*n*Log[Log[n]]/(2*Log[n]^2)],{n,2,100}]

a(n) = floor(prime(n)-n*log(n) + n - n*log(log(n)) - (n/log(n))*(log(log(n)) - 2) + (log(log(n)) - 6)*n*log(log(n))/(2*log(n)^2)); \\ Michel Marcus, Feb 22 2025

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))

A250622 a(n) = floor(n*log(prime(n)))-prime(n), n >= 1.

Original entry on oeis.org

-2, -1, -1, 0, 0, 2, 2, 4, 5, 4, 6, 6, 7, 9, 10, 10, 10, 12, 12, 14, 17, 17, 18, 18, 17, 18, 22, 23, 27, 28, 23, 25, 25, 28, 26, 29, 30, 30, 32, 33, 33, 37, 34, 38, 40, 44, 40, 36, 38, 42, 45, 45, 49, 47, 48, 49, 49, 53, 54, 57, 61, 59, 53, 56, 60, 63, 57, 58, 56, 60, 63, 64, 64, 65, 66, 69, 70, 69

Offset: 1

Freimut Marschner, Dec 02 2014

Since n*log(prime(n)) > prime(n), n >= 4 and ceiling(prime(n) - n*log(n)) < prime(n), then n*log(n) < prime(n) < n*log(prime(n)), n >= 4. This inequality is included in the prime number theorem PNT. Remark: a(n) >= 0 for n >=4 otherwise a(n) < 0.

			n = 1, a(1) = floor(1*0.6931...) - 2 = 0 - 2 = -2;
n = 5, a(5) = floor(5*2.3978...) - 11 = floor( 11.9894...) - 11 = 11 - 11 = 0;
n = 6, a(6) = floor(6*2.5649...) - 13 = floor(15.3896...) - 13 = 15 - 13 = 2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040, A064658 (ceiling(prime(n) - n*log(n))), A250621 (floor(n*log(prime(n)))).

a250622[n_Integer] := Table[Floor[i*Log[Prime[i]]] - Prime[i], {i, n}]; a250622[121] (* Michael De Vlieger, Dec 11 2014 *)

vector(100,n,floor(n*log(prime(n))-prime(n))) \\ Derek Orr, Dec 13 2014

a(n) = floor(n*log(prime(n))) - prime(n) = A250621(n) - A000040(n).

cp's OEIS Frontend

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

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A064659 a(n) = round(prime(n) - n*log(n)).

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A272231 a(n) = round(n / pi(n)), where pi(n) is the prime-counting function.

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A204325 Residual of an asymptotic formula for the n-th prime: a(n) = floor(prime(n)-n*log(n) + n - n*log(log(n)) - (n/log(n))*(log(log(n)) - 2) + (log(log(n)) - 6)*n*log(log(n))/(2*log(n)^2)).

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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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A250622 a(n) = floor(n*log(prime(n)))-prime(n), n >= 1.

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A204325 Residual of an asymptotic formula for the n-th prime: a(n) = floor(prime(n)-nlog(n) + n - nlog(log(n)) - (n/log(n))(log(log(n)) - 2) + (log(log(n)) - 6)nlog(log(n))/(2log(n)^2)).

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).