A104272 -id:A104272 - cp's OEIS frontend

A189026 There appear to be at least n primes in the range (x-sqrt(x), x] for all x >= a(n).

127, 1367, 2531, 2539, 6007, 7457, 10061, 10847, 23531, 35797, 35801, 38557, 44497, 47111, 69767, 69809, 88321, 107687, 110419, 110431, 113723, 127217, 250673, 250681, 250687, 250703, 268487, 268493, 286381, 286393, 302563, 302567, 360947, 369821, 405199

Offset: 1

T. D. Noe, Apr 15 2011

nonn

These terms exist only if a strong form of Oppermann's conjecture that for any k>1 there is a prime between k^2-k and k^2 is true. Note that every term is prime. Sequence A189024 gives the number of primes in the range (x-sqrt(x), x]. The index of the prime a(n), that is, primepi(a(n)), is approximately (2.4*n)^2. These primes are generated in a manner similar to the Ramanujan primes (A104272).

T. D. Noe, Table of n, a(n) for n = 1..1000
Wikipedia, Oppermann's conjecture

Cf. A189025, A189027.

A189027 There appear to be at least n primes in the range (x-2*sqrt(x), x] for all x >= a(n).

Original entry on oeis.org

2, 3, 37, 139, 331, 1409, 1423, 1427, 2239, 3163, 3181, 3511, 6547, 7433, 7457, 7487, 10061, 11777, 11779, 14401, 18899, 19081, 19373, 23537, 24763, 27617, 27673, 32027, 32051, 38113, 43573, 43579, 47269, 47279, 50839, 61463, 88643, 88651, 88657, 88729

Offset: 1

T. D. Noe, Apr 15 2011

nonn

These terms exist only if a strong form of Legendre's conjecture that there is a prime between consecutive squares is true. Note that every term is prime. Sequence A189025 gives the number of primes in the range (x-2*sqrt(x), x]. The index of prime a(n), that is, primepi(a(n)), is approximately (5n)^2. These primes are generated in a manner similar to the Ramanujan primes (A104272).

T. D. Noe, Table of n, a(n) for n = 1..1000
Wikipedia, Legendre's conjecture

Cf. A189024, A189026.

A190502 Number of Ramanujan primes <= 2^n.

Original entry on oeis.org

0, 1, 1, 1, 2, 4, 7, 13, 23, 42, 75, 137, 255, 463, 872, 1612, 3030, 5706, 10749, 20387, 38635, 73584, 140336, 268216, 513705, 985818, 1894120, 3645744, 7027290, 13561906, 26207278, 50697533, 98182656, 190335585, 369323301, 717267167, 1394192236, 2712103833

Offset: 0

John W. Nicholson, May 11 2011

nonn

Dana Jacobsen, Table of n, a(n) for n = 0..56

Cf. A007053, A036378, A104272, A181671, A190501.

\\ With RR[.] is a list of A104272(.). The output of this program is n, a(n), and RR[a(n)].
j=0; while(2^jJohn W. Nicholson, Dec 01 2012

use ntheory ":all"; sub a190502 { scalar(@{ramanujan_primes(1 << shift)}) } say a190502($) for 0..20; # _Dana Jacobsen, Dec 19 2015

use ntheory ":all"; my $t = 0; for my $e (1..32) { $t += scalar(@{ramanujan_primes(2**($e-1)+1,2**$e)}); say "$e $t" } # Dana Jacobsen, Dec 19 2015

use ntheory ":all"; say ramanujan_prime_count(2**$) for 0..47; # _Dana Jacobsen, Jan 03 2016

Extended by T. D. Noe, May 11 2011
Extended to n = 32 by John W. Nicholson, Dec 01 2012
a(33)-a(41) from Dana Jacobsen, Dec 19 2015

A191225 Number of Ramanujan primes R_k between triangular numbers T(n-1) < R_k <= T(n).

Original entry on oeis.org

0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 2, 0, 2, 1, 1, 2, 1, 2, 0, 2, 3, 2, 1, 2, 2, 2, 1, 4, 2, 2, 1, 0, 4, 3, 5, 1, 3, 2, 1, 5, 1, 2, 3, 4, 4, 4, 2, 2, 2, 4, 2, 3, 4, 3, 5, 4, 3, 2, 5, 4, 2, 5, 1, 6, 1, 5, 5, 7, 2, 2, 1, 10, 6, 6, 2, 2, 5, 0, 3, 7, 5, 4, 6, 7, 4

Offset: 1

John W. Nicholson, May 27 2011

nonn

The function eta(x), A191228, returns the greatest value of k of R_k <= x, and where R_k is the k-th Ramanujan prime (A104272).

			Write the numbers 1, 2, ... in a triangle with n terms in the n-th row; a(n) = number of Ramanujan primes in n-th row.
Triangle begins
1                 (0 Ramanujan primes, eta(1) = 0)
2  3              (1 Ramanujan primes, eta(3) - eta(1) = 1)
4  5  6           (0 Ramanujan primes, eta(6) - eta(3) = 0)
7  8  9  10       (0 Ramanujan primes, eta(10) - eta(6) = 0)
11 12 13 14 15    (1 Ramanujan primes, eta(15) - eta(10) = 1)
16 17 18 19 20 21 (1 Ramanujan primes, eta(21) - eta(15) = 1)

T. D. Noe, Table of n, a(n) for n = 1..10000
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. A000217, A000040, A104272, A191228, A014085, A190661, A083382, A191226, A191227.

terms = 100; nn = terms^2; 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[3 nn]}];
A104272 = R + 1;
eta = Table[Boole[MemberQ[A104272, k]], {k, 1, nn}] // Accumulate;
T[n_] := n(n+1)/2;
a[1] = 0; a[n_] := eta[[T[n]]] - eta[[T[n-1]]];
Array[a, terms] (* Jean-François Alcover, Nov 07 2018, using T. D. Noe's code for A104272 *)

use ntheory ":all"; sub a191225 { my $n = shift; ramanujan_prime_count( (($n-1)*$n)/2+1, ($n*($n+1))/2 ); } say a191225($) for 1..10; # _Dana Jacobsen, Dec 30 2015

a(n) = eta(T(n))- eta(T(n-1)).

A193880 0.75-Ramanujan primes R_{0.75,n}: a(n) is the smallest number such that for all x >= a(n), we have pi(x) - pi(0.75x) >= n, where pi(x) is the number of primes <= x.

Original entry on oeis.org

11, 29, 59, 67, 101, 149, 157, 163, 191, 227, 269, 271, 307, 379, 383, 419, 431, 433, 443, 457, 563, 593, 601, 641, 643, 673, 701, 709, 733, 827, 829, 907, 937, 947, 971, 1019, 1033, 1039, 1051, 1087, 1187, 1193, 1217, 1277, 1427, 1429, 1433, 1481, 1483, 1487

Offset: 1

Nadine Amersi, Olivia Beckwith (obeckwith(AT)gmail.com), Steven J. Miller (Steven.J.Miller(AT)williams.edu), Ryan Ronan (ronan2(AT)cooper.edu), Jonathan Sondow, Aug 07 2011

nonn

See comment to A193761. - Vladimir Shevelev, Aug 18 2011

See additional comments and links in A290394. - Jonathan Sondow, Aug 01 2017

			a(1) = A290394(3) = 11. - _Jonathan Sondow_, Aug 01 2017

N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, Generalized Ramanujan primes, arXiv:1108.0475 [math.NT], 2011.
N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, Generalized Ramanujan primes, Combinatorial and Additive Number Theory, Springer Proc. in Math. & Stat., CANT 2011 and 2012, Vol. 101 (2014), 1-13
V. Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes, arXiv:0909.0715 [math.NT], 2009-2011.
V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4

Cf. A104272 (Ramanujan primes), A193761 (0.25-Ramanujan primes), A164952, A290394 (first (1 + 1/n)-Ramanujan prime).

a(n) >= A104272(n).

A205301 Last occurrence of n partitions in A204814.

Original entry on oeis.org

1312, 1501, 2446, 2776, 4381, 3676, 4951, 6541, 5686, 7771, 8326, 7696, 8011, 10471, 9256, 9871, 11041, 11626, 15256, 12751, 15511, 15151, 14956, 19441, 16801, 20596, 20101, 22291, 17611, 17341, 18451, 21856, 19051, 21226, 23761, 20842, 24796, 22651, 22546

Offset: 0

John W. Nicholson, Jan 27 2012

nonn

Conjectured lower bound to be increasing for increasing n.

Related to Goldbach's conjecture.

			1312 is the last observed 0 so for n=0, a(0)=1312.

Donovan Johnson, Table of n, a(n) for n = 0..1000 (from search done for first 25*10^6 terms in A204814)

Cf. A104272, A173634, A204814.

a(10)-a(38) from Donovan Johnson, Jan 28 2012

A212541 Let p_n=prime(n), n>=1. Then a(n) is the maximal prime p which differs from p_n, for which the intervals (p/2,p_n/2), (p,p_n], if pp_n, contain the same number of primes, and a(n)=0, if no such prime p exists.

Original entry on oeis.org

0, 11, 11, 11, 7, 17, 13, 29, 29, 23, 41, 41, 37, 47, 43, 59, 53, 67, 61, 0, 97, 97, 97, 97, 89, 0, 107, 103, 127, 149, 109, 149, 149, 151, 137, 139, 167, 167, 163, 179, 173, 0, 227, 229, 229, 233, 229, 227, 223, 211, 199, 0, 0, 263, 263, 257, 0, 281, 281

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 20 2012

nonn

a(n)A104272).

a(n)=0 if and only if p_n is a peculiar prime, i.e., simultaneously Ramanujan and Labos (A080359) prime (see sequence A164554).

			Let n=4, p_n=7. Since 7 is not Ramanujan prime, then a(4) = A104272(4-pi(3.5)) = A104272(2) = 11.

V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4

Cf. A212493, A104272, A080359, A164554.

If p_n is not a Ramanujan prime, then a(n) = A104272(n-pi(p_n/2)).

A214924 Number of primes <= A214756(n).

Original entry on oeis.org

1, 1, 1, 7, 20, 28, 96, 152, 185, 212, 1179, 1829, 2217, 3382, 14350, 30780, 31528, 40929, 103498, 104047, 149674, 325845, 1094396, 1319933, 2850163, 6957867, 10539421, 10655453

Offset: 1

John W. Nicholson, Jul 29 2012

nonn

a(n) = pi(A214756(n)).

			A214756(5) = 71, so a(5) = primepi(A214756(5)) = primepi(71) = 20.

Cf. A000101, A104272, A214756, A001223.

Cf. A002386, A214925, A214926, A214757.

a(n) = A000217(A214756(n))

Extension to a(28) added by John W. Nicholson, Nov 11 2013

A214925 Number of primes <= A214757(n).

Original entry on oeis.org

5, 5, 5, 10, 25, 31, 104, 159, 190, 219, 1186, 1832, 2227, 3388, 14358, 30804, 31547, 40935, 103522, 104072, 149690, 325853, 1094426, 1319950, 2850175, 6957880, 10539433, 10655464

Offset: 1

John W. Nicholson, Aug 06 2012

			A214757(4) = 29, so a(4) = primepi(A214757(4)) = primepi(29) = 10.

Cf. A000101, A104272, A214756, A001223.

Cf. A002386, A214924, A214926, A214757.

a(n) = pi(A214757(n)) = A000217(A214757(n)).

Extension to a(28) added by John W. Nicholson, Nov 11 2013

A214926 Difference A214925(n) - A214924(n), prime count between Ramanujan primes bounding maximal gap primes.

Original entry on oeis.org

4, 4, 4, 3, 5, 3, 8, 7, 5, 7, 7, 3, 10, 6, 8, 24, 19, 6, 24, 25, 16, 8, 30, 17, 12, 13, 12, 11

Offset: 1

John W. Nicholson, Aug 06 2012

nonn

Conjecture: For every n > 0, a(n) > 1.

Let rho(m) = A179196(m), for any n, let m be an integer such that p_(rho(m)) <= p_n and p_(n+1) <= p_(rho(m+1)), then rho(m) <= n < n + 1 <= rho(m + 1), therefore A001223(n) = p_(n+1) - p_n <= p_rho(m+1) - p_rho(m) = A182873(m). For all rho(m) = A179196(m), A001223(rho(m)) < A165959(m). (Comment copied from A001223). John W. Nicholson, Nov 17 2013

			a(4) = pi(A214757(4)) - pi(A214756(4)) = 10 - 7 = 3

Cf. A214924, A214925, A214756, A214757.

Cf. A000101, A104272, A001223, A002386.

a(n) = pi(A214757(n)) - pi(A214756(n)).

a(n) = rho(A214757(n)) - rho(A214756(n)).

Extension to a(28) added by John W. Nicholson, Nov 11 2013

cp's OEIS Frontend

A189026 There appear to be at least n primes in the range (x-sqrt(x), x] for all x >= a(n).

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A189027 There appear to be at least n primes in the range (x-2*sqrt(x), x] for all x >= a(n).

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A190502 Number of Ramanujan primes <= 2^n.

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A191225 Number of Ramanujan primes R_k between triangular numbers T(n-1) < R_k <= T(n).

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A193880 0.75-Ramanujan primes R_{0.75,n}: a(n) is the smallest number such that for all x >= a(n), we have pi(x) - pi(0.75x) >= n, where pi(x) is the number of primes <= x.

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A205301 Last occurrence of n partitions in A204814.

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A212541 Let p_n=prime(n), n>=1. Then a(n) is the maximal prime p which differs from p_n, for which the intervals (p/2,p_n/2), (p,p_n], if pp_n, contain the same number of primes, and a(n)=0, if no such prime p exists.

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A214924 Number of primes <= A214756(n).

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A214925 Number of primes <= A214757(n).

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