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

Author

Cino Hilliard, Dec 21 2006

Keywords

• nonn
• uned

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

Examples

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

Crossrefs

Cf. A007097.

Programs

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 *)

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

}

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.

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