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)).

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

Author

José María Grau Ribas, Jan 14 2012

Keywords

• sign

Comments

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

References

• 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.

Links

• 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

Crossrefs

Cf. A064658, A059111.

Programs

Mathematica

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}]

PARI

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