This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A124627 #36 Dec 07 2019 17:22:09
%S A124627 2,3,5,11,33,127,715,5345,52692,648344,9737826,174442666,3657513487,
%T A124627 88362834417,2428095525614,75063691591379,2586559741900744,
%U A124627 98552043877145945,4123221751454999891,188272405177875090033,9332039515886416792536,499720579610294249596689,28785866289101759323472435,1776891233143817540293248652
%N A124627 Riemann-Gram approximation to A007097(n+1) using A007097(n).
%C A124627 The largest presently [as of Dec 2006] known value of prime(10^n) is
%C A124627 prime(10^18) = 44211790234832169331 this compares to
%C A124627 primex(10^18) = 44211790234127235727 accurate to 11 places
%C A124627 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:
%C A124627 n prime(10^n) primex(10^n) rel. error
%C A124627 -- -------------------- -------------------- ------------
%C A124627 6 15485863 15484040 1.1772 E-4
%C A124627 7 179424673 179431239 -3.6594 E-4
%C A124627 8 2038074743 2038076587 -9.0478 E-5
%C A124627 9 22801763489 22801797576 -1.4949 E-5
%C A124627 10 252097800623 252097715777 3.3655 E-6
%C A124627 11 2760727302517 2760727752353 -1.6294 E-6
%C A124627 12 29996224275833 29996225393465 -3.7259 E-7
%C A124627 13 323780508946331 323780512411510 -1.0702 E-7
%C A124627 14 3475385758524527 3475385760290723 -5.0820 E-8
%C A124627 15 37124508045065437 37124508056355511 -3.0411 E-9
%C A124627 16 394906913903735329 394906913798224969 2.6718 E-9
%C A124627 17 4185296581467695669 4185296581676470048 -4.9883 E-11
%C A124627 18 44211790234832169331 44211790234127235727 1.5944 E-11
%F A124627 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.
%e A124627 A007097(17) = 75063692618249;
%e A124627 Primex(75063692618249) = 2586559741900744;
%e A124627 A007097(18) = 2586559730396077;
%e A124627 Primex(2586559730396077) = 98552043877145945;
%e A124627 A007097(19) ~ 98552043800000000.
%t A124627 RiemannGram[x_] := Module[{n = 1, L, s = 1, r}, L = r = Log[x];
%t A124627 While[s < 10^30 r, s = s + r/(Zeta[n + 1] n); n++; r = r L/n]; s];
%t A124627 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];
%t A124627 Primex /@ NestList[Prime, 1, 15] (* _Birkas Gyorgy_, Apr 04 2011 *)
%o A124627 (PARI) xeqprimex(n) = {
%o A124627 my(a,x); a = [1, 2, 3, 5, 11, 31, 127, 709, 5381, 52711, 648391, 9737333, 174440041, 3657500101, 88362852307, 2428095424619, 75063692618249, 2586559730396077];
%o A124627 for(x=1,n, print1(round(primex(a[x]))",") ) }
%o A124627 \\ Approximates the n-th prime number to an accuracy of log10(n)/2 places.
%o A124627 primex(n) = {
%o A124627 my(x,px,r1,r2,r,p10,b,e,est);
%o A124627 if(n==1,return(2)); \\ force to 2
%o A124627 b=10; \\ Select base
%o A124627 p10=log(n)/log(10); \\ Determine p10 = power of 10 of n to adjust b^p10
%o A124627 if(Rg(b^p10*log(b^(p10+1)))< b^p10,m=p10+1,m=p10);
%o A124627 r1 = 0; r2 = 7.718281828; \\ Real kicker. if r2=1, it fails at 1e117
%o A124627 for(x=1, 100,
%o A124627 r=(r1+r2)/2;
%o A124627 est = (b^p10*log(b^(m+r)));
%o A124627 px = Rg(est);
%o A124627 if(px <= b^p10,r1=r,r2=r); r=(r1+r2)/2; );
%o A124627 est;
%o A124627 }
%o A124627 Rg(x) = \\ Gram's Riemann Approx of Pi(x)
%o A124627 { my(n=1,L,s=1,r);
%o A124627 L=r=log(x);
%o A124627 while(s<10^40*r, s=s+r/zeta(n+1)/n; n=n+1; r=r*L/n);
%o A124627 (s)
%o A124627 }
%Y A124627 Cf. A007097.
%K A124627 nonn,uned
%O A124627 1,1
%A A124627 _Cino Hilliard_, Dec 21 2006
%E A124627 a(19) and a(20) found by _David Baugh_ using a program by Xavier Gourdon and _Andrey V. Kulsha_, Oct 25 2007
%E A124627 a(21), a(22) and a(23) calculated by _David Baugh_, Feb 10 2015
%E A124627 a(24) calculated by _David Baugh_, May 16 2016