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 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

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Last modified October 15 00:03 EDT 2019. Contains 328025 sequences. (Running on oeis4.)