login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A124627 Riemann-Gram approximation to A007097(n+1) using A007097(n). 0
2, 3, 5, 11, 33, 127, 715, 5345, 52692, 648344, 9737826, 174442666, 3657513487, 88362834417, 2428095525614, 75063691591379, 2586559741900744, 98552043877145945, 4123221751454999891, 188272405177875090033, 9332039515886416792536, 499720579610294249596689, 28785866289101759323472435, 1776891233143817540293248652 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The largest presently 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), relative 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 log10(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: A173422 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 25 04:50 EDT 2019. Contains 323539 sequences. (Running on oeis4.)