A104272 -id:A104272 - cp's OEIS frontend

A220293 Chebyshev numbers C_2(n): a(n) is the smallest number such that if x >= a(n), then theta(x) - theta(x/2) >= n*log(x), where theta(x) = sum_{prime p <= x} log p.

11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127, 149, 151, 167, 179, 223, 229, 233, 239, 241, 263, 269, 281, 307, 311, 347, 349, 367, 373, 401, 409, 419, 431, 433, 443, 461, 487, 503, 569, 571, 587, 593, 599, 601, 607, 641, 643, 647, 653

Offset: 1

Vladimir Shevelev, Charles R Greathouse IV and Peter J. C. Moses, Dec 09 2012

nonn

Up to a(100)=1489, only two terms of the sequence (a(17)=223 and a(36)=443) are not Ramanujan numbers (A104272), and the sequence is missing only the following Ramanujan numbers up to 1489: 2, 181, 227, 439, 491, 1283, and 1301. The latter observation shows how closely the ratio theta(x)/log(x) approximates the number of primes <= x (i.e., pi(x)).
A generalization: for a real number v>1, the v-Chebyshev number C_v(n) is the smallest integer k such that if x>=k, then theta(x)-theta(x/v)>=n*log x. In particular, a(n)=C_2(n). For another example, if v=4/3, then, at least up to 3319, all (4/3)-Chebyshev numbers are (4/3)-Ramanujan primes as in Shevelev's link (cf. A193880, where c=1/v=3/4 is excepted), and in this case the sequence is missing only the following (4/3)-Ramanujan numbers up to 3319: 11 and 1567.
Like Chebyshev numbers, all v-Chebyshev numbers are primes.

N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and J. Sondow, Generalized Ramanujan primes, arXiv 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, J. Integer Seq. 15 (2012) Article 12.5.4
Vladimir Shevelev, Charles R. Greathouse IV, and Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv:1212.2785

Cf. A104272.

For n >= 2, A104272(n) <= a(n-1) <= prime(3n).

A225907 Smallest n-Ramanujan prime that is less than half of the next n-Ramanujan prime, or 0 if none exists.

Original entry on oeis.org

0, 2, 11, 41, 587, 14143

Offset: 0

Jonathan Sondow, Jun 08 2013

In A192824 Noe defines 0-Ramanujan primes to be simply primes, and 1-Ramanujan primes to be Ramanujan primes. Define the k-th 2-Ramanujan prime to be the smallest number R'_k (the notation in Paksoy 2012) with the property that the interval (x/2,x] contains at least k 1-Ramanujan primes, for any x >= R'_k. Continuing inductively, define n-Ramanujan primes in terms of (n-1)-Ramanujan primes.
Only the first three terms 0, 2, 11 are proved (by Chebyshev, Ramanujan, and Paksoy, respectively). The rest are conjectural--see the 2nd comment in A192821.
See A104272 for additional comments, references, links, and cross-refs.
Is it true that for every n there exists K = K(n) such that for all k > K, the k-th n-Ramanujan prime is greater than half of the (k+1)-th n-Ramanujan prime? (Equivalently, is there a largest n-Ramanujan prime that is less than half of the next n-Ramanujan prime?) It is true for n = 0 by Bertrand's Postulate (see A062234), and for n = 1 by a theorem of Paksoy. Is it even true that if n is fixed, then (k-th n-Ramanujan prime) ~ ((k+1)-th n-Ramanujan prime) as k -> infinity? - Jonathan Sondow, Dec 16 2013

			By Bertrand's Postulate (proved by Chebyshev), prime(k+1) < 2*prime(k) for all k, so a(0) = 0.
Ramanujan proved that the Ramanujan primes begin 2, 11, ..., so a(1) = 2.
Paksoy proved that the 2-Ramanujan primes begin 11, 41,..., so a(2) = 11.
It appears that the 3-Ramanujan primes begin 41, 149, ...; if true, then a(3) = 41.
It appears that the 4-Ramanujan primes begin 569, 571, 587, 1367 ...; if true, then a(4) = 587.

Murat Baris Paksoy, Derived Ramanujan primes: R'_n, arXiv:1210.6991 [math.NT], 2012.

Cf. A000040 (0-Ramanujan primes), A104272 (1-Ramanujan primes), A192820 (2-Ramanujan primes), A192821 (3-Ramanujan primes), A192822 (4-Ramanujan primes), A192823 (5-Ramanujan primes), A192824 (least n-Ramanujan prime). Cf. also A233822 = 2*R(n) - R(n+1) and A062234.

A228592 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 prime powers p^k <= x (k >= 1).

Original entry on oeis.org

2, 3, 7, 11, 23, 27, 29, 41, 59, 67, 71, 79, 101, 107, 109, 125, 127, 149, 167, 169, 179, 181, 227, 229, 233, 239, 263, 269, 281, 283, 307, 311, 347, 349, 359, 367, 373, 401, 409, 419, 431, 433, 439, 487, 491, 503, 521, 569, 587, 593, 599, 601, 607, 617, 641

Offset: 1

Vladimir Shevelev, Aug 27 2013

nonn

It is a subsequence of A000961.

Peter J. C. Moses, Table of n, a(n) for n = 1..10000

Cf. A000961, A104272, A228520.

a(n) <= A228520(n); a(n) ~ A000040(2*n), when n goes to the infinity.

More terms from Peter J. C. Moses, Aug 27 2013

A233822 a(n) = 2*R(n) - R(n+1), where R(n) is the n-th Ramanujan prime.

Original entry on oeis.org

-7, 5, 5, 17, 35, 35, 51, 63, 45, 93, 95, 87, 105, 147, 135, 155, 177, 135, 225, 225, 227, 237, 219, 257, 257, 255, 303, 275, 345, 331, 361, 345, 393, 399, 407, 429, 427, 417, 435, 483, 479, 437, 567, 555, 581, 587, 597, 595, 573, 639, 639, 641, 647

Offset: 1

Jonathan Sondow, Dec 16 2013

sign

a(n) = 2*A104272(n) - A104272(n+1).
Paksoy proved that a(n) > 0 for n > 1.
Paksoy's theorem is the analog for Ramanujan primes of Chebychev's theorem (Bertrand's postulate) that 2*prime(n) - prime(n+1) > 0 for n > 0 (see A062234).

			The only negative term is a(1) = 2*R(1) - R(2) = 2*2 - 11 = -7.

Dana Jacobsen, Table of n, a(n) for n = 1..10000
Baris Paksoy, Derived Ramanujan primes: R'_n, arXiv:1210.6991 [math.NT], 2012.

Cf. A062234, A104272, A225907.

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[3*nn]}]; R = R + 1;
Most[2 R - RotateLeft[R]] (* Jean-François Alcover, Dec 06 2018, after T. D. Noe in A104272 *)

use ntheory ":all"; say 2*nth_ramanujan_prime($)-nth_ramanujan_prime($+1) for 1..10 # Dana Jacobsen, Sep 02 2017

A088634 Index of the first occurrence of n in A066888.

Original entry on oeis.org

1, 3, 2, 9, 17, 28, 30, 36, 41, 54, 74, 51, 65, 92, 100, 112, 118, 132, 108, 154, 158, 161, 172, 175, 210, 197, 215, 255, 248, 239, 236, 316, 297, 291, 340, 321, 330, 345, 334, 400, 406, 402, 423, 394, 445, 452, 509, 493, 507, 481, 526, 546, 561, 584, 565, 598

Offset: 0

Amarnath Murthy, Oct 26 2003

For all 0 < n < 1000, a(n) < A104272(n). - John W. Nicholson, May 22 2011

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A066888, A000217, A000040, A190661.

nn=100; k=0; t=Table[0,{nn}]; cnt=0; While[cntT. D. Noe, May 19 2011 *)

More terms from David Wasserman, Aug 16 2005

A185005 Ramanujan primes R_(3,2)(n): a(n) is the smallest number such that if x >= a(n), then pi_(3,2)(x) - pi_(3,2)(x/2) >= n, where pi_(3,2)(x) is the number of primes==2 (mod 3) <= x.

Original entry on oeis.org

11, 23, 47, 59, 83, 107, 131, 167, 227, 233, 239, 251, 263, 281, 347, 383, 401, 419, 431, 443, 479, 563, 587, 593, 641, 647, 653, 659, 719, 743, 809, 821, 839, 863, 941, 947, 971, 1019, 1049, 1061, 1091, 1151, 1187, 1217, 1223, 1259, 1283

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Dec 18 2012

nonn

All terms are primes==2 (mod 3).
For the definition of generalized Ramanujan numbers, see Section 6 of the Shevelev, Greathouse, & Moses link.
We conjecture that for all n >= 1, a(n) <= A104272(3*n). This conjecture is based on observation that, if interval (x/2, x] contains >= 3*n primes, then at least n of them are of the form 3*k+2.

Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv:1212.2785

Cf. A104272, A185004, A185006, A185007.

Table[1 + NestWhile[#1 - 1 &, A104272[[3 k]], Count[Mod[Select[Range@@{Floor[#1/2 + 1], #1}, PrimeQ], 3], 2] >= k &], {k, 1, 10}]

lim(a(n)/prime(4*n)) = 1 as n tends to infinity.

A185006 Ramanujan primes R_(4,1)(n): a(n) is the smallest number such that if x >= a(n), then pi_(4,1)(x) - pi_(4,1)(x/2) >= n, where pi_(4,1)(x) is the number of primes==1 (mod 4) <= x.

Original entry on oeis.org

13, 37, 41, 89, 97, 109, 149, 229, 233, 241, 257, 277, 281, 317, 349, 397, 401, 409, 421, 433, 569, 593, 601, 641, 653, 661, 709, 757, 761, 821, 929, 937, 941, 953, 977, 997, 1009, 1021, 1049, 1061, 1093, 1097, 1117, 1193, 1213, 1237, 1249

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Dec 18 2012

nonn

All terms are primes==1 (mod 4).
A general conception of generalized Ramanujan numbers, see in Section 6 of the Shevelev, Greathouse IV, & Moses link.
We conjecture that for all n >= 1, a(n) <= A104272(3*n). This conjecture is based on observation that, if interval (x/2, x] contains >= 3*n primes, then at least n of them are of the form 4*k+1.

Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv:1212.2785

Cf. A104272, A185005, A185004, A185007.

Table[1 + NestWhile[#1 - 1 &, A104272[[3 k]], Count[Mod[Select[Range@@{Floor[#1/2 + 1], #1}, PrimeQ], 4], 1] >= k &], {k, 1, 10}] using the code nn = 1000; A104272 = Table[0, {nn}]; s = 0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s < nn, A104272[[s + 1]] = k], {k, Prime[3*nn]}]; A104272 = A104272 + 1 (* T. D. Noe, Nov 15 2010 *)

lim(a(n)/prime(4*n)) = 1 as n tends to infinity.

A190501 Number of Ramanujan primes R_k such that 2^(n-1) < R_k <= 2^n.

Original entry on oeis.org

0, 1, 0, 0, 1, 2, 3, 6, 10, 19, 33, 62, 118, 208, 409, 740, 1418, 2676, 5043, 9638, 18248, 34949, 66752, 127880, 245489, 472113, 908302, 1751624, 3381546, 6534616, 12645372, 24490255, 47485123, 92152929, 178987716, 347943866, 676925069, 1317911597, 2567659990, 5005877954, 9765539069, 19062301793, 37230980158, 72756216207, 142253989491, 278275735952, 544621563320, 1066382258001

Offset: 0

John W. Nicholson, May 11 2011

nonn

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

Cf. A007053, A036378, A104272, A190502.

\\ With RR[.] is a list of A104272(.). The output of this program is (for n>=1) n, A190502(n), and RR[a(n)], a(n).
j=0;while(2^j

use ntheory ":all"; say "$ ",ramanujan_prime_count(1 << $) - ramanujan_prime_count(1 << ($-1)) for 0..56; # _Dana Jacobsen, May 05 2017

Extended by T. D. Noe, May 11 2011
Modified the name as to match offset to A190502 and added leading term, John W. Nicholson, May 12 2011
Extended to n = 32 by John W. Nicholson, Dec 01 2012
Extended to n = 47, using A190502 data, by John W. Nicholson, Jan 31 2016

A191226 First occurrence of number n of Ramanujan primes in A191225.

Original entry on oeis.org

1, 2, 12, 22, 29, 36, 65, 69, 117, 118, 73, 100, 108, 154, 161, 200, 254, 172, 274, 239, 340, 321, 334, 330, 345, 471, 378, 481, 357, 526, 522, 515, 610, 635, 612, 655, 648, 792, 809, 731, 797, 594, 806, 851, 988, 886, 963, 933, 1005, 1111, 927, 1124, 970

Offset: 0

John W. Nicholson, May 28 2011

nonn

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
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, A191225, A191227.

A191227 Last known occurrence of number n of Ramanujan primes in A191225.

Original entry on oeis.org

79, 194, 153, 284, 420, 333, 454, 592, 504, 412, 652, 512, 486, 617, 613, 660, 1130, 753, 1002, 849, 1060, 957, 1034, 1037, 1198, 961, 969, 1056, 1368, 1400, 1264, 1314, 1301, 1683, 1510, 1571, 1532, 1625, 1771, 1810, 1745, 1907, 1961, 1877, 1851, 2104, 2097

Offset: 0

John W. Nicholson, May 28 2011

nonn

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, A191225, A191226.

cp's OEIS Frontend

A220293 Chebyshev numbers C_2(n): a(n) is the smallest number such that if x >= a(n), then theta(x) - theta(x/2) >= n*log(x), where theta(x) = sum_{prime p <= x} log p.

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A225907 Smallest n-Ramanujan prime that is less than half of the next n-Ramanujan prime, or 0 if none exists.

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A228592 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 prime powers p^k <= x (k >= 1).

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A233822 a(n) = 2*R(n) - R(n+1), where R(n) is the n-th Ramanujan prime.

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Perl

A088634 Index of the first occurrence of n in A066888.

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A185005 Ramanujan primes R_(3,2)(n): a(n) is the smallest number such that if x >= a(n), then pi_(3,2)(x) - pi_(3,2)(x/2) >= n, where pi_(3,2)(x) is the number of primes==2 (mod 3) <= x.

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A185006 Ramanujan primes R_(4,1)(n): a(n) is the smallest number such that if x >= a(n), then pi_(4,1)(x) - pi_(4,1)(x/2) >= n, where pi_(4,1)(x) is the number of primes==1 (mod 4) <= x.

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A190501 Number of Ramanujan primes R_k such that 2^(n-1) < R_k <= 2^n.

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PARI

Perl

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A191226 First occurrence of number n of Ramanujan primes in A191225.

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A228592 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 prime powers p^k <= x (k >= 1).