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 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 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

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