The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”). Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A190124 Decimal expansion of Ramanujan prime constant: Sum_{n>=1} (1/R_n)^2, where R_n is the n-th Ramanujan prime, A104272(n). 3
 2, 6, 5, 5, 6, 3, 2, 7, 5, 8, 0 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS By computing all Ramanujan primes less than 10^9, we find that about 9 decimal places of the sum should be correct: 0.265563275 (truncated, not rounded). The following table shows the number of Ramanujan primes between powers of 10 and the sum of the squared reciprocals of those primes. 1          1    0.25000000000000000 2          9    0.01477600368240514 3         62    0.00072814919125266 4        487    0.00005457480850461 5       3900    0.00000417097012694 6      32501    0.00000034491619098 7     279106    0.00000002943077197 8    2444255    0.00000000255829675 9   21731345    0.00000000022619762 Total:          0.26556327578374667 - T. D. Noe, May 05 2011 From Jonathan Sondow, May 06 2011: (Start) Since R_n > n, the bound Sum_{n > N} 1/(R_n)^2 < 1/N holds, by the integral test. Taking N = #{R_n < 10^9} = 24491666, the error is < 4.09 x 10^-8. Using the stronger inequality R_n > 2n log 2n (from "Ramanujan primes and Bertrand's postulate"), the error is actually < 2.94 * 10^-11. So the sum 0.265563275... is correct. The next digit is either 7 or 8. (End) A190124 and A085548 (Prime Zeta(2)) converge by comparison with A013661 (Zeta(2)), which converges by the integral test. As real numbers, A190124 < A085548 < A013661. - Robert G. Wilson v, May 08 2011 Prime Zeta(2) - (this constant) = 0.4522474200 - 0.2655632757 = 0.186684144 (truncated, not rounded). - John W. Nicholson, May 24 2011 From Dana Jacobsen, Jul 27 2015: (Start) Calculating more Ramanujan primes, we can expand on the earlier table, which should give us more terms.    1            1  0.25000000000000000000  0.25000000000000000000    2            9  0.26477600368240513652  0.01477600368240513652    3           62  0.26550415287365779725  0.00072814919125266073    4          487  0.26555872768216240627  0.00005457480850460902    5         3900  0.26556289865228934691  0.00000417097012694064    6        32501  0.26556324356848032844  0.00000034491619098153    7       279106  0.26556327299925229431  0.00000002943077196587    8      2444255  0.26556327555754904279  0.00000000255829674847    9     21731345  0.26556327578374665897  0.00000000022619761618   10    195606622  0.26556327580402332096  0.00000000002027666198   11   1778301947  0.26556327580586060071  0.00000000000183727975   12  16301375641  0.26556327580602856045  0.00000000000016795974. (End) LINKS J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly 116 (2009), 630-635. J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011. EXAMPLE 0.265563275... PROG (Perl) use ntheory ":all"; use Math::MPFR qw/Rmpfr_get_str Rmpfr_set_default_prec Rmpfr_printf/; Rmpfr_set_default_prec(500); my \$limit = shift || 9; my(\$maxexp, \$sum) = (9, Math::MPFR->new(0)); for my \$e (1..\$limit) {   my(\$numrp, \$psum) = (0, Math::MPFR->new(0));   if (\$e <= \$maxexp) {     my \$rp = ramanujan_primes(10**(\$e-1), 10**\$e);     \$numrp += scalar @\$rp;     \$psum += (1/Math::MPFR->new("\$_"))**2 for @\$rp;   } else {     for my \$k (10**(\$e-\$maxexp-1) .. 10**(\$e-\$maxexp)-1) {       my \$rp = ramanujan_primes(\$k*10**\$maxexp, (\$k+1)*10**\$maxexp);       \$numrp += scalar @\$rp;       \$psum += (1/Math::MPFR->new("\$_"))**2 for @\$rp;     }   }   Rmpfr_printf("%2d ", \$e);   Rmpfr_printf("%14lu   ", \$numrp);   Rmpfr_printf("%.20Rf  ", \$sum += \$psum);   Rmpfr_printf("%.20Rf\n", \$psum); } # Dana Jacobsen, Jul 27 2015 CROSSREFS Cf. A078437, A085548, A104272. Sequence in context: A309040 A316134 A273621 * A067548 A245698 A053793 Adjacent sequences:  A190121 A190122 A190123 * A190125 A190126 A190127 KEYWORD nonn,cons,more AUTHOR John W. Nicholson, May 04 2011 EXTENSIONS a(10) and a(11) (from data above by Dana Jacobsen_, Jul 27 2015) added by John W. Nicholson, Dec 17 2015 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.

Last modified December 2 17:41 EST 2021. Contains 349445 sequences. (Running on oeis4.)