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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

Table of n, a(n) for n=0..10.

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