A104272 -id:A104272 - cp's OEIS frontend

A194217 a(n) = A104272(n)-A080359(n).

0, 8, 4, 10, 10, 4, 6, 6, 0, 24, 0, 4, 18, 36, 12, 10, 6, 0, 36, 36, 34, 0, 0, 12, 0, 10, 24, 18, 34, 0, 14, 0, 22, 0, 0, 10, 0, 0, 18, 24, 0, 4, 60, 48, 10, 0, 0, 0, 0, 28, 24, 0, 0, 0, 16, 36, 36, 6, 8, 12, 36, 10, 0, 0, 24, 0, 22, 54, 30, 0, 14, 12, 18, 22

Offset: 1

Vladimir Shevelev, Aug 18 2011

nonn

Conjecture: Asymptotic density of nonzero terms is 3/4.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
V. Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes
J. Sondow, Ramanujan Prime. (MathWorld)

Cf. A104272, A080359, A193507, A194184, A194186.

nn = 100;
R = Table[0, {nn}]; s = 0;
Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s < nn, R[[s+1]] = k], {k, Prime[3nn]}
];
A104272 = R = R + 1;
T = Table[0, {nn + 1}]; s = 0;
Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s <= nn && T[[s+1]] == 0, T[[s+1]] = k], {k, Prime[3nn]}
];
A080359 = Rest[T];
A104272 - A080359 (* Jean-François Alcover, Aug 19 2018, after T. D. Noe *)

A162996 a(n) = Round(kn * (log(kn)+1)), with k = 2.216 as an approximation of R_n = n-th Ramanujan prime A104272(n) and Abs(a(n)-R_n) < 2 * Sqrt(a(n)) for n in [1..1000].

Original entry on oeis.org

4, 11, 19, 28, 38, 48, 58, 69, 80, 91, 102, 114, 126, 138, 150, 162, 174, 187, 200, 212, 225, 238, 251, 265, 278, 291, 305, 318, 332, 345, 359, 373, 387, 401, 415, 429, 443, 458, 472, 486, 501, 515, 530, 544, 559, 573, 588, 603, 618, 632, 647, 662, 677, 692

Offset: 1

Daniel Forgues, Jul 21 2009, Jul 29 2009

nonn

a(n) approximates the {kn}-th prime number, which in turn approximates the n-th Ramanujan prime, and k = 2.216 is nearly optimal for n in [1..1000] since a(n) - 2*sqrt(a(n)) < R_n < a(n) + 2*sqrt(a(n)) in that range. Since R_n ~ Prime(2n) ~ 2n * (log(2n)+1) ~ 2n * log(2n), whereas A162996(n) ~ Prime(kn) ~ kn * (log(kn)+1) ~ kn * log(kn), giving A162996(n) / R_n ~ k/2 = 2.216/2 = 1.108 which implies an asymptotic overestimate of about 10% (a better approximation would need k to depend on n and be asymptotic to 2).

R_n is the smallest number such that if x >= R_n, then pi(x) - pi(x/2) >= n, where pi(x) is the number of primes <= x.

Daniel Forgues, Table of n, a(n) for n=1..1000

Cf. A163160 (Round(kn * (log(kn)+1)) - R_n, where k = 2.216 and R_n = n-th Ramanujan prime).

Cf. A104272 (Ramanujan primes).

A163160 a(n) = A162996(n) - R_n = round(kn * (log(kn)+1)) - R_n, with k = 2.216 and R_n = n-th Ramanujan Prime A104272(n) and where Abs(a(n)) < 2 * sqrt(A162996(n)) for n in [1..1000].

Original entry on oeis.org

2, 0, 2, -1, -3, 1, -1, 2, 9, -6, 1, 7, -1, -11, -1, -5, -5, 6, -27, -17, -8, -1, 10, 2, 9, 10, -2, 7, -15, -4, -8, 0, -14, -8, -4, -2, 10, 19, 11, -1, 10, 12, -39, -27, -28, -20, -11, 2, 11, -9, 4, 15, 24, 33, 30, 3, 11, 14, 17, 14, -11, -7, 6, 18, 7, 18, 10, -31, -19, -9, -14

Offset: 1

Daniel Forgues, Jul 21 2009, Jul 29 2009

sign

A162996(n) approximates the {kn}-th prime number, which in turn approximates the n-th Ramanujan prime, with k = 2.216 nearly optimal for n in [1..1000] since a(n) - 2*sqrt(a(n)) < R_n < a(n) + 2*sqrt(a(n)) in that range. Since R_n ~ Prime(2n) ~ 2n * (log(2n)+1) ~ 2n * log(2n), whereas A162996(n) ~ Prime(kn) ~ kn * (log(kn)+1) ~ kn * log(kn), giving A162996(n) / R_n ~ k/2 = 2.216/2 = 1.108 which implies an asymptotic overestimate of about 10.8% (a better approximation would need k to depend on n and be asymptotic to 2). Consequently, a(n) - 2*sqrt(a(n)) < R_n < a(n) + 2*sqrt(a(n)) will fail pretty early (R_n falling below the lower bound) as n grows beyond 1000.

Daniel Forgues, Table of n, a(n) for n=1..1000

Cf. A162996 (Round(kn * (log(kn)+1)), with k = 2.216 as an approximation of R_n = n-th Ramanujan Prime).

Cf. A104272 (Ramanujan primes: a(n) is the smallest number such that if x >= a(n), then pi(x) - pi(x/2) >= n, where pi(x) is the number of primes <= x).

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).

Original entry on oeis.org

2, 6, 5, 5, 6, 3, 2, 7, 5, 8, 0

Offset: 0

John W. Nicholson, May 04 2011

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    0.25000000000000000
        9    0.01477600368240514
       62    0.00072814919125266
      487    0.00005457480850461
     3900    0.00000417097012694
    32501    0.00000034491619098
   279106    0.00000002943077197
  2444255    0.00000000255829675
 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  0.25000000000000000000  0.25000000000000000000
          9  0.26477600368240513652  0.01477600368240513652
         62  0.26550415287365779725  0.00072814919125266073
        487  0.26555872768216240627  0.00005457480850460902
       3900  0.26556289865228934691  0.00000417097012694064
      32501  0.26556324356848032844  0.00000034491619098153
     279106  0.26556327299925229431  0.00000002943077196587
    2444255  0.26556327555754904279  0.00000000255829674847
   21731345  0.26556327578374665897  0.00000000022619761618
  195606622  0.26556327580402332096  0.00000000002027666198
 1778301947  0.26556327580586060071  0.00000000000183727975
16301375641  0.26556327580602856045  0.00000000000016795974. (End)

			0.265563275...

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.

Cf. A078437, A085548, A104272.

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

a(10) and a(11) (from data above by Dana Jacobsen_, Jul 27 2015) added by John W. Nicholson, Dec 17 2015

A190413 primepi(R_{nm}) <= nprimepi(R_m) for m >= a(n), where R_k is the k-th Ramanujan prime (A104272).

Original entry on oeis.org

1, 1245, 189, 189, 85, 85, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

T. D. Noe, May 11 2011

This is Conjecture 1 in the paper by Sondow, Nicholson, and Noe. The conjecture has been verified for n <= 20 and Ramanujan primes less than 10^9.
A restatement is rho(n*m) <= n*rho(m) for m >= a(n), where rho = A179196.
The conjecture has been proven for n > 10^300 by Shichun Yang and Alain Togbé. - Jonathan Sondow, Jan 21 2016
The conjecture has been proven for n > 38 and m > 9 by Christian Axler. Complete exception list can be found in remark of paper. - John W. Nicholson, Aug 04 2019

Christian Axler, On the number of primes up to the n-th Ramanujan prime, arXiv:1711.04588 [math.NT], 2017.
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, J. Integer Seq. 14 (2011) Article 11.6.2.
Shichun Yang and Alain Togbé, On the estimates of the upper and lower bounds of Ramanujan primes, Ramanujan J., online 14 August 2015, 1-11.

Cf. A007395, A104272, A179196, A190414.

For all n >= 20, a(n) = 2.

A190414 primepi(R_m) <= iprimepi(R_j) for any factorization m=ij if j >= a(i), where R_k is the k-th Ramanujan prime (A104272).

Original entry on oeis.org

1, 2490, 567, 756, 425, 510, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100

Offset: 1

John W. Nicholson, May 10 2011

nonn

This is another interpretation of Conjecture 1 in the paper by Sondow, Nicholson, and Noe. The conjecture has been verified for i <= 20 and Ramanujan primes less than 10^9.

The conjecture has been proven for i > 38 and j > 9 by Christian Axler. Complete exception list can be found in remark of paper. - John W. Nicholson, Aug 04 2019

Christian Axler, On the number of primes up to the n-th Ramanujan prime, arXiv:1711.04588 [math.NT], 2017.
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2.
Shichun Yang and Alain Togbé, On the estimates of the upper and lower bounds of Ramanujan primes, Ramanujan J., online 14 August 2015, 1-11.

Cf. A104272, A179196, A190413.

For all n >= 20, a(n) = 2*n.

A173081 Number of twin prime pairs < 10^n that contain at least one Ramanujan prime (A104272).

Original entry on oeis.org

0, 6, 28, 167, 964, 6305, 45082, 335919, 2605867, 20841010, 170395131

Offset: 1

T. D. Noe, Nov 22 2010

It appears that this gives the number of Ramanujan primes < 10^n that are the lesser prime in a twin prime pair. Equivalently, this sequence also gives the number of Ramanujan primes p with p+2 also prime less than 10^n.
It appears that no upper twin prime is a Ramanujan prime without the corresponding lower twin prime also being a Ramanujan prime.
This is proved in Section 4 of "Ramanujan Primes: Bounds, Runs, Twins, and Gaps".

J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2.

Cf. A178128 (Ramanujan primes p such that p+2 is prime), A007508 (number of twin primes pairs < 10^n), A181678 (number of twin Ramanujan primes pairs < 10^n).

nn=50000; t=Table[0,{nn}]; s=0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s

a(10)-a(11) from Dana Jacobsen, Apr 29 2015

A182391 Numbers n for which A104272(n) = A080359(n).

Original entry on oeis.org

1, 9, 11, 18, 22, 23, 25, 30, 32, 34, 35, 37, 38, 41, 46, 47, 48, 49, 52, 53, 54, 63, 64, 66, 70, 75, 76, 79, 80, 82, 84, 94, 98, 99, 101, 102, 105, 108, 109, 110, 113, 114, 115, 124, 127, 128, 131, 135, 136, 139, 140, 148, 149, 150, 151, 154, 156, 158, 160

Offset: 1

Vladimir Shevelev, Apr 27 2012

nonn

Number m is in the sequence iff 1) there exists only composite number k such that 2*k-1 is prime and A060715(k)=m; 2) there is no prime p such that 2*p-1 is prime and A060715(p)=m-1.

Cf. A104272, A080359, A164554, A193507, A194184, A194186, A194217, A182366.

A194217(n)=0.

A190303 Decimal expansion of sum of alternating series of reciprocals of Ramanujan primes, Sum_{n>=1} (1/R_n)(-1)^(n-1), where R_n is the n-th Ramanujan prime, A104272(n).

Original entry on oeis.org

4, 4, 6, 6, 8, 4, 3, 0, 7

Offset: 0

John W. Nicholson, May 07 2011

Computed 0.446684 for n = 1 to 65536, using Open Office Calc. Next digit expected to be between 2 and 3.
By computing all Ramanujan primes less than 10^9, we find that about 9 decimal places of the sum should be correct: 0.446684307 (truncated, not rounded). The following table shows the number of Ramanujan primes between powers of 10 and the sum of the alternating reciprocals of those primes.
1          1      0.50000000000000000
2          9     -0.05765566386047510
3         62      0.00388002010130731
4        487      0.00050881775862179
5       3900     -0.00004384563815649
6      32501     -0.00000552572415587
7     279106      0.00000045427780897
8    2444255      0.00000005495474474
9   21731345     -0.00000000549864067
Total:            0.44668430669928564 - T. D. Noe, May 08 2011
Let E_n denote the error after the first n terms in the series. Then by the Alternating Series Test, 1/R_{n+1} - 1/R_{n+2} < E_n < 1/R_{n+1}. [Jonathan Sondow, May 10 2011]

			0.446684307...

J. Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009-2010.
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, J. Integer Seq. 14 (2011) Article 11.6.2
Eric W. Weisstein, Prime Sums

Cf. A104272, A085548, A078437, A190124.

Sum_{n>=1} (-1)^(n-1)(1/R_n), where R_n is the n-th Ramanujan prime, A104272(n).

Definition corrected by Jonathan Sondow, May 10 2011

A190881 Exception list of where A190661(n) < A104272(n) for n > 0.

Original entry on oeis.org

43, 45, 68, 93, 145, 341, 655, 3177, 3383, 3424, 4696, 5109, 5116, 5183, 5201, 5225, 7195, 7574

Offset: 1

John W. Nicholson, May 23 2011

nonn

a(n) is the value of n of A190661 and A104272 at the exception.

Cf. A066888, A000217, A000040, A088634, A190661, A104272.

Corrected data for the changed offset and data of A190661 - John W. Nicholson, Nov 20 2013

cp's OEIS Frontend

A194217 a(n) = A104272(n)-A080359(n).

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A162996 a(n) = Round(kn * (log(kn)+1)), with k = 2.216 as an approximation of R_n = n-th Ramanujan prime A104272(n) and Abs(a(n)-R_n) < 2 * Sqrt(a(n)) for n in [1..1000].

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A163160 a(n) = A162996(n) - R_n = round(kn * (log(kn)+1)) - R_n, with k = 2.216 and R_n = n-th Ramanujan Prime A104272(n) and where Abs(a(n)) < 2 * sqrt(A162996(n)) for n in [1..1000].

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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).

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A190413 primepi(R_{n*m}) <= n*primepi(R_m) for m >= a(n), where R_k is the k-th Ramanujan prime (A104272).

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A190414 primepi(R_m) <= i*primepi(R_j) for any factorization m=i*j if j >= a(i), where R_k is the k-th Ramanujan prime (A104272).

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A173081 Number of twin prime pairs < 10^n that contain at least one Ramanujan prime (A104272).

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Mathematica

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A182391 Numbers n for which A104272(n) = A080359(n).

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A190303 Decimal expansion of sum of alternating series of reciprocals of Ramanujan primes, Sum_{n>=1} (1/R_n)(-1)^(n-1), where R_n is the n-th Ramanujan prime, A104272(n).

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A190881 Exception list of where A190661(n) < A104272(n) for n > 0.

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A190413 primepi(R_{nm}) <= nprimepi(R_m) for m >= a(n), where R_k is the k-th Ramanujan prime (A104272).

A190414 primepi(R_m) <= iprimepi(R_j) for any factorization m=ij if j >= a(i), where R_k is the k-th Ramanujan prime (A104272).