cp's OEIS Frontend

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)).

%I A204325 #38 Feb 22 2025 12:04:51
%S A204325 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,
%T A204325 12,9,14,10,11,12,11,11,12,9,13,10,8,5,11,18,16,12,11,11,8,12,12,12,
%U A204325 13,9,9,7,4,8,16,14,10,8,16,16,20,16
%N 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)).
%C A204325 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).
%C A204325 The first negative term is a(214) = -2. - _Jason Yuen_, Feb 17 2025
%D A204325 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.
%H A204325 Michel Marcus, <a href="/A204325/b204325.txt">Table of n, a(n) for n = 2..10000</a>
%H A204325 Pierre Dusart, <a href="http://arxiv.org/abs/1002.0442">Estimates of Some Functions Over Primes without R.H.</a>, arXiv:1002.0442 [math.NT], 2010.
%H A204325 Wikipedia, <a href="http://en.wikipedia.org/wiki/Prime_number_theorem">Prime number theorem</a>
%t A204325 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}]
%o A204325 (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
%Y A204325 Cf. A064658, A059111.
%K A204325 sign
%O A204325 2,1
%A A204325 _José María Grau Ribas_, Jan 14 2012