A124627 Riemann-Gram approximation to A007097(n+1) using A007097(n).

2, 3, 5, 11, 33, 127, 715, 5345, 52692, 648344, 9737826, 174442666, 3657513487, 88362834417, 2428095525614, 75063691591379, 2586559741900744, 98552043877145945, 4123221751454999891, 188272405177875090033, 9332039515886416792536, 499720579610294249596689, 28785866289101759323472435, 1776891233143817540293248652

Offset

1,1

Comments

The largest presently [as of Dec 2006] known value of prime(10^n) is

prime(10^18) = 44211790234832169331 this compares to

primex(10^18) = 44211790234127235727 accurate to 11 places

Here the sign of prime(x)-primex(x) is positive. However, the sign changes as x varies. The following is a table with the relative error and sign change:

n prime(10^n) primex(10^n) rel. error

-- -------------------- -------------------- ------------

6 15485863 15484040 1.1772 E-4

7 179424673 179431239 -3.6594 E-4

8 2038074743 2038076587 -9.0478 E-5

9 22801763489 22801797576 -1.4949 E-5

10 252097800623 252097715777 3.3655 E-6

11 2760727302517 2760727752353 -1.6294 E-6

12 29996224275833 29996225393465 -3.7259 E-7

13 323780508946331 323780512411510 -1.0702 E-7

14 3475385758524527 3475385760290723 -5.0820 E-8

15 37124508045065437 37124508056355511 -3.0411 E-9

16 394906913903735329 394906913798224969 2.6718 E-9

17 4185296581467695669 4185296581676470048 -4.9883 E-11

18 44211790234832169331 44211790234127235727 1.5944 E-11

Links

Table of n, a(n) for n=1..24.

Formula

Primex(n) ~ prime(n). Prime(n) is the n-th prime number. Primex(n) is the Riemann-Gram approximation of Prime(n) accurate to log_10(n)/2 + 1 digits for large n. The sequence is primex(A007097(n)) for n = 1 to 18.

Example

A007097(17) = 75063692618249;
Primex(75063692618249) = 2586559741900744;
A007097(18) = 2586559730396077;
Primex(2586559730396077) = 98552043877145945;
A007097(19) ~ 98552043800000000.

Mathematica

RiemannGram[x_] := Module[{n = 1, L, s = 1, r}, L = r = Log[x];
   While[s < 10^30 r, s = s + r/(Zeta[n + 1] n); n++; r = r L/n]; s];
Primex[n_] :=  Module[{r1, r2, r, est},   If[n == 1, r = 2, r1 = n Log[n]; r2 = 2 r1;    For[i = 1, i < 50, i++, r = (r1 + r2)/2; est = RiemannGram[r]; If[est < n, r1 = r, r2 = r]]]; Round@r];
Primex /@ NestList[Prime, 1, 15] (* Birkas Gyorgy, Apr 04 2011 *)

Prog

(PARI) xeqprimex(n) = {
my(a, x); a = [1, 2, 3, 5, 11, 31, 127, 709, 5381, 52711, 648391, 9737333, 174440041, 3657500101, 88362852307, 2428095424619, 75063692618249, 2586559730396077];
for(x=1, n, print1(round(primex(a[x]))", ") ) }
\\ Approximates the n-th prime number to an accuracy of log10(n)/2 places.
primex(n) = {
my(x, px, r1, r2, r, p10, b, e, est);
if(n==1, return(2)); \\ force to 2
b=10; \\ Select base
p10=log(n)/log(10); \\ Determine p10 = power of 10 of n to adjust b^p10
if(Rg(b^p10*log(b^(p10+1)))< b^p10, m=p10+1, m=p10);
r1 = 0; r2 = 7.718281828; \\ Real kicker. if r2=1, it fails at 1e117
for(x=1, 100,
   r=(r1+r2)/2;
   est = (b^p10*log(b^(m+r)));
   px = Rg(est);
   if(px <= b^p10, r1=r, r2=r); r=(r1+r2)/2; );
  est;
}
Rg(x) = \\ Gram's Riemann Approx of Pi(x)
{ my(n=1, L, s=1, r);
L=r=log(x);
while(s<10^40*r, s=s+r/zeta(n+1)/n; n=n+1; r=r*L/n);
(s)
}

Crossrefs

Cf. A007097.

Sequence in context: A375554 A132745 A124538 * A305971 A064095 A061935

Adjacent sequences: A124624 A124625 A124626 * A124628 A124629 A124630

Keyword

nonn,uned

Author

Cino Hilliard, Dec 21 2006

Extensions

a(19) and a(20) found by David Baugh using a program by Xavier Gourdon and Andrey V. Kulsha, Oct 25 2007

a(21), a(22) and a(23) calculated by David Baugh, Feb 10 2015

a(24) calculated by David Baugh, May 16 2016

Status

approved