A104272 -id:A104272 - cp's OEIS frontend

A084139 a(n) is the largest number for which exactly n primes are bounded between a(n) and 2a(n) exclusively.

1, 5, 8, 14, 20, 23, 29, 33, 35, 48, 50, 53, 63, 74, 75, 83, 89, 90, 113, 114, 116, 119, 120, 131, 134, 140, 153, 155, 173, 174, 183, 186, 200, 204, 209, 215, 216, 219, 230, 243, 245, 251, 284, 285, 293, 296, 299, 300, 303, 320, 321, 323, 326, 329, 338, 359, 363

Offset: 0

Harry J. Smith, May 15 2003

nonn

a(n) is the index of last occurrence of n in A060715. This calculation relies on the fact that Pi(2*m)-Pi(m) > m/(3*Log(m)) for m>=5. It can be shown that every integer >= 0 occurs in A060715, so there is no problem in finding the last occurrence.

A168421(n) = nextprime(a(n)), where nextprime(x) is the next prime > x. Note: some a(n) may be prime, therefore nextprime(x) not equal to x. - John W. Nicholson, Oct 11 2013

			a(10) = 50 since ten primes last arise between 50 and 100: 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 140.

T. D. Noe, Table of n, a(n) for n = 0..1000
Eric W. Weisstein, MathWorld: Bertrand's Postulate

Cf. A060715, A060756, A084138, A084140, A084141, A084142.

nn = 100; t = Table[0, {nn}]; Do[m = PrimePi[2*n] - PrimePi[n]; If[0 < m <= nn, t[[m]] = n], {n, 15*nn}]; Join[{1}, t] (* T. D. Noe, Dec 31 2012 *)

a(n) = floor((A104272(n)+1)/2) for n >= 1. - John W. Nicholson, Oct 11 2013

a(n) = A084140(n+1) - 1. - John W. Nicholson, Oct 11 2013

A164952 Primes p with the property: if p/3 is in the interval (p_m, p_(m+1)), where p_m>=3 and p_k is the k-th prime, then the interval (p, 3p_(m+1)) contains a prime.

Original entry on oeis.org

2, 3, 11, 17, 23, 29, 41, 43, 59, 61, 71, 73, 79, 97, 101, 103, 107, 131, 137, 149, 151, 163, 167, 179, 191, 193, 223, 227, 229, 239, 251, 257, 269, 271, 277, 281, 311, 331, 347, 349, 353, 359, 367, 373, 383, 397, 419, 421, 431, 433, 439, 457, 461, 463, 479, 491

Offset: 1

Vladimir Shevelev, Sep 01 2009

nonn

For k>1 (not necessarily integer), we call a Ramanujan k-prime R_n^(k) the prime a_k(n) which is the smallest number such that if x >= a_k(n), then pi(x)- pi(x/k) >= n. Note that, the sequence of all primes corresponds to the case of "k=oo". These numbers possess the following properties: R_n^(k)~p_((k/(k-1))n) as n tends to the infinity; if A_k(x) is the counting function of the Ramanujan k-primes not exceeding x, then A_k(x)~(1-1/k)\pi(x) as x tends to the infinity; let p be a Ramanujan k-prime, such that p/k is in the interval (p_m, p_(m+1)), where p_m>=3 and p_n is the n-th prime. Then the interval (p, k*p_(m+1)) contains a prime. Conjecture. For every k>=2 there exist non-Ramanujan k-primes, which possess the latter property. For example, for k=2, the smallest such prime is 109. Problem. For every k>2 to estimate the smallest non-Ramanujan k-prime which possesses the latter property. [From Vladimir Shevelev, Sep 01 2009]

All Ramanujan 3-primes are in the sequence.

			If p=61, the p/3 is in the interval (19, 23); we see that the interval (61,69) contains a prime (67).

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, A164368, A164288.

nn=1000; t=Table[0, {nn}]; s=0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/3], s--]; If[s

Extended and edited by T. D. Noe, Nov 22 2010

Comments edited by Jonathan Sondow, Aug 27 2011

A174635 Prime numbers that are not Ramanujan primes.

Original entry on oeis.org

3, 5, 7, 13, 19, 23, 31, 37, 43, 53, 61, 73, 79, 83, 89, 103, 109, 113, 131, 137, 139, 157, 163, 173, 191, 193, 197, 199, 211, 223, 251, 257, 271, 277, 283, 293, 313, 317, 331, 337, 353, 359, 379, 383, 389, 397, 421, 443, 449, 457, 463, 467, 479, 499, 509, 521

Offset: 1

T. D. Noe, Nov 29 2010

nonn

Complement of A104272 in the primes. Not the same as A059788.

Also known as non-Ramanujan Primes. - John W. Nicholson, Jan 29 2012

Donovan Johnson, Table of n, a(n) for n = 1..10000
J. Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009-201; Amer. Math. Monthly 116 (2009), 630-635. - _John W. Nicholson_, Jan 29 2012
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2 - _John W. Nicholson_, Jan 29 2012.

Cf. A104272.

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;
Complement[Prime[Range[PrimePi[Last[R]]]], R] (* Jean-François Alcover, Nov 05 2018, after T. D. Noe in A104272 *)

use ntheory ":all";  my @n = grep { !is_ramanujan_prime($) } @{primes(1e3)}; say "[@n]"; # _Dana Jacobsen, Jul 15 2016

use ntheory ":all"; my %r; $r{$} = 1 for @{ramanujan_primes(1e7)}; say for grep { !exists $r{$} } @{primes(1e7)}; # Dana Jacobsen, Jul 15 2016

A190661 a(n) is the least number m such that there are at least n primes in the range (T(k-1), T(k)] for all k >= m, where T(k) is the k-th triangular number.

Original entry on oeis.org

1, 7, 16, 33, 52, 66, 79, 72, 109, 93, 121, 119, 130, 153, 169, 194, 180, 222, 235, 275, 294, 267, 256, 296, 329, 339, 333, 420, 383, 373, 372, 454, 396, 443, 449, 504, 463, 574, 559, 512, 592, 602, 596, 541, 652, 585, 683, 656, 687, 689, 708

Offset: 0

John W. Nicholson, May 18 2011

nonn

All values and even whether the sequence is well defined are conjectural.
a(n) is the conjectured index of the last occurrence of n in A066888.
It is conjectured that for every n >= 0, a(n) > n.
With R_n the n-th Ramanujan prime (A104272), it is conjectured that for every n >= 0, (1/2) R_n <= a(n) < (20/13) R_n. These bounds have been verified for all n up to 8000. For most n <= 8000, we have a(n) > R_n, with exceptions listed in A190881.

			Because it appears that A066888(7) = 1 is the last 1 of that sequence, a(1) = 7.

T. D. Noe, Table of n, a(n) for n = 0..8000

Cf. A066888, A000217, A000040, A088634, A104272, A190881.

Edited by T. D. Noe, May 19 2011

A192820 2-Ramanujan primes: the interval (x/2,x] has at least n Ramanujan primes for x >= a(n) but not for x = a(n) - 1.

Original entry on oeis.org

11, 41, 59, 97, 149, 151, 227, 229, 233, 239, 263, 307, 367, 373, 401, 409, 569, 571, 587, 593, 599, 641, 643, 647, 653, 719, 751, 821, 937, 941, 1009, 1019, 1021, 1031, 1049, 1051, 1061, 1063, 1217, 1367, 1373, 1423, 1427, 1439, 1481, 1487, 1549, 1553, 1559

Offset: 1

T. D. Noe, Jul 11 2011

nonn

It is conjectured that primepi(a(n)) <= 7*n for all n. - T. D. Noe, Aug 26 2011
Subsequence of the Ramanujan primes A104272, by the minimality of a(n). - Jonathan Sondow, Aug 21 2012
Paksoy (2012) denotes a(n) by R'n and calls it "the n-th derived Ramanujan prime." He proves the bounds on R'_n below. - _Jonathan Sondow, Oct 29 2012

T. D. Noe, Table of n, a(n) for n = 1..1000
M. B. Paksoy, Derived Ramanujan primes: R'_n, arXiv:1210.6991 [math.NT], 2012.

Cf. A104272 (Ramanujan primes), A192821, A192822, A192823, A192824, A225907.

R(2n) <= a(n) < R(3n), where R(n) = the n-th Ramanujan prime (Paksoy 2012).
p(4n)  < a(n) < p(9n), where p(n) = the n-th prime (Paksoy 2012).
a(n) < p(8n) for n >= 5315 (Paksoy 2012).
R(2n) ~ a(n) ~ p(4n) as n -> oo (Paksoy 2012).

Definition clarified by Jonathan Sondow, Aug 21 2012

A080360 a(n) is the largest positive integer x such that the number of unitary-prime-divisors of x! equals n. Same as the largest positive integer x such that the number of primes in (x/2,x] equals n.

Original entry on oeis.org

10, 16, 28, 40, 46, 58, 66, 70, 96, 100, 106, 126, 148, 150, 166, 178, 180, 226, 228, 232, 238, 240, 262, 268, 280, 306, 310, 346, 348, 366, 372, 400, 408, 418, 430, 432, 438, 460, 486, 490, 502, 568, 570, 586, 592, 598, 600, 606, 640, 642, 646, 652, 658, 676

Offset: 1

Labos Elemer, Feb 21 2003

nonn

			n=5: in 46! five unitary-prime-divisors[UPD] appear: {29,31,37,41,43}. In larger factorials number of UPD is not more equal 5. Thus a(5)=46.

S. Ramanujan, Collected Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, S. Aiyar, P. Venkatesvara and B. M. Wilson), Amer. Math. Soc., Providence, 2000, pp. 208-209.

Amiram Eldar, Table of n, a(n) for n = 1..10000
S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.
V. Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes, arXiv:0909.0715 [math.NT], 2009-2011.
J. Sondow, Ramanujan Prime in MathWorld
J. Sondow and E. W. Weisstein, Bertrand's Postulate in MathWorld
J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, 116 (2009), 630-635; arXiv:0907.5232 [math.NT], 2009-2010.
Wikipedia, Ramanujan prime

Cf. A056171, A056169, A000720, A000142, A080359.

Cf. A104272 (Ramanujan primes).

nn = 60; 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]}];
Rest[R] (* Jean-François Alcover, Dec 02 2018, after T. D. Noe in A104272 *)

a(n) = Max{x; Pi[x]-Pi[x/2]=n} = Max{x; A056171(x)=n} = Max{x; A056169(n!)=n}; where Pi()=A000720().

a(n) = A104272(n+1) - 1. [Jonathan Sondow, Aug 11 2008]

Definition corrected by Jonathan Sondow, Aug 10 2008

A164371 The smaller members of twin prime pairs which are non-Ramanujan primes.

Original entry on oeis.org

3, 5, 137, 191, 197, 521, 617, 1319, 1619, 1697, 1931, 1949, 2111, 2549, 2591, 2969, 3119, 3329, 3389, 3557, 3929, 4157, 4217, 4271, 4517, 5279, 5417, 7331, 7349, 7757, 8009, 8429, 8837, 9011, 9281, 9437, 9857, 9929, 10007, 10037, 10457, 10529, 10859

Offset: 1

Vladimir Shevelev, Aug 14 2009, Aug 17 2009

nonn

The terms greater than 5 of the sequence are in A164288.

Dana Jacobsen, 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. A104272, A001359, A164288, A164554

nn = 1000;
A001359 = Select[Prime[Range[2 nn]], PrimeQ[# + 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;
Complement[A001359, A104272] (* Jean-François Alcover, Oct 28 2018, after T. D. Noe in A104272 *)

use ntheory ":all"; for (@{twin_primes(1e4)}) { say if !is_ramanujan_prime($) } # _Dana Jacobsen, Dec 30 2015

A001359 \ A104272. - R. J. Mathar, Sep 17 2009

Extended by R. J. Mathar, Sep 17 2009

A174602 Smallest prime that begins a run of n Ramanujan primes that are consecutive primes.

Original entry on oeis.org

2, 67, 227, 227, 227, 2657, 2657, 2657, 2657, 2657, 2657, 2657, 2657, 562871, 793487, 809707, 809707, 984241, 984241, 984241, 6234619, 11652013, 41662651, 41662651, 41662651, 94653397, 383825567, 869730887, 953913871, 953913871, 953913871

Offset: 1

T. D. Noe, Nov 29 2010

nonn

The first run of 13 consecutive Ramanujan primes was mentioned by Sondow.

Starting at index m = A191228(a(n)) in A190874(m), the first instance of a count of n - 1 consecutive 1's is seen. - John W. Nicholson, Dec 15 2011

			67 and 71 are the first two Ramanujan primes that are consecutive primes, so a(2) = 67.

Dana Jacobsen, Table of n, a(n) for n = 1..42
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, arXiv:1105.2249 [math.NT], 2011.
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. A104272 (Ramanujan primes), A174641 (runs of non-Ramanujan primes).

nn=10000; t=Table[0, {nn}]; len=Prime[3*nn]; s=0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s

use ntheory ":all"; my $r=ramanujan_primes(1e8); my $max = 0; for (0..$#$r-2) { my $k=0; $k++ while next_prime($r->[$+$k]) == $r->[$+$k+1]; say ++$max," ",$r->[$] while $k >= $max; } # _Dana Jacobsen, Jul 14 2016

A178127 Lesser of twin Ramanujan primes.

Original entry on oeis.org

149, 179, 227, 239, 347, 431, 569, 599, 641, 821, 1019, 1049, 1061, 1427, 1487, 1607, 1787, 1997, 2081, 2129, 2237, 2267, 2657, 2687, 2711, 2789, 2999, 3167, 3257, 3299, 3359, 3527, 3539, 3581, 3671, 3917, 4091, 4127, 4229, 4241, 4337, 4547, 4637, 4649

Offset: 1

Jonathan Sondow, May 20 2010

nonn

By definition, a number p is a member if p and p+2 are Ramanujan primes A104272.
Conjecture: For all n > 570, more than 1/4 of the twin prime pairs < n are both Ramanujan primes.
Motivation for the conjecture is in "Ramanujan primes and Bertrand's postulate" Section 7.
Subsequence of A178128.
See A001359 and A104272 for additional comments, links, and references.

			a(1) = 149 because (149,151) is the 1st pair of twin primes both of which are Ramanujan primes.
11 is not a member even though 11 and 13 are twin primes and 11 is a Ramanujan prime, because 13 is not also a Ramanujan prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..674 from R. J. Mathar)
B. Ghusayni, Subsets of prime numbers, Int. J. Math. Comp. Sci. 7 (2) 2012
Jonathan Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009-2010.
Jonathan Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, 116 (2009), 630-635.
Jonathan 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. A001359, A104272, A164371, A178128, A190654.

Cf. A181678 (number of twin Ramanujan prime pairs less than 10^n).

n := 1:
for i from 1 do
    if A104272(i+1) = A104272(i)+2 then
        printf("%d %d\n",n,A104272(i)) ;
        n := n+1 ;
    end if;
end do: # produces b-file, R. J. Mathar, Sep 21 2017

nn = 1000; 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;
twins1 = Position[A104272 // Differences, 2] // Flatten;
A104272[[twins1]] (* Jean-François Alcover, Oct 28 2018, after T. D. Noe in A104272 *)

use ntheory ":all"; my $r = ramanujan_primes(1e5); my @rt = @$r[grep { $r->[$+1]-$r->[$]==2 } 0..$#$r-1]; say "@rt"; # Dana Jacobsen, Sep 06 2015

{A104272(n): A104272(n+1) = A104272(n)+2}.

a(n) = A190654(2n-1) = A190654(2n) - 2.

A182365 The largest prime in some interval of the form (2prime(k),2prime(k+1)) if this interval contains at least 2 primes.

Original entry on oeis.org

13, 19, 31, 43, 53, 61, 73, 103, 113, 131, 139, 157, 173, 193, 199, 251, 271, 293, 313, 353, 379, 421, 443, 463, 499, 509, 523, 577, 613, 619, 661, 691, 733, 743, 757, 773, 811, 829, 859, 883, 911, 953, 971, 997, 1013, 1039, 1069, 1093, 1109, 1123, 1153

Offset: 1

Vladimir Shevelev, Apr 26 2012

nonn

These are called "left primes" in A166251.

			For k=6 we have 2*13 < 29 < 31 < 2*17, and the interval contains two primes. Therefore 31 is in the sequence.

Cf. A166252, A166251, A164368, A104272, A080359, A164333, A164288, A164294, A164554, A166307.

n = 0; t = {}; While[Length[t] < 100, n++; ps = Select[Range[2*Prime[n], 2*Prime[n + 1]], PrimeQ]; If[Length[ps] >= 2, AppendTo[t, ps[[-1]]]]]; t (* T. D. Noe, Apr 30 2012 *)

cp's OEIS Frontend

A084139 a(n) is the largest number for which exactly n primes are bounded between a(n) and 2a(n) exclusively.

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A164952 Primes p with the property: if p/3 is in the interval (p_m, p_(m+1)), where p_m>=3 and p_k is the k-th prime, then the interval (p, 3p_(m+1)) contains a prime.

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A174635 Prime numbers that are not Ramanujan primes.

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Perl

A190661 a(n) is the least number m such that there are at least n primes in the range (T(k-1), T(k)] for all k >= m, where T(k) is the k-th triangular number.

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A192820 2-Ramanujan primes: the interval (x/2,x] has at least n Ramanujan primes for x >= a(n) but not for x = a(n) - 1.

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A080360 a(n) is the largest positive integer x such that the number of unitary-prime-divisors of x! equals n. Same as the largest positive integer x such that the number of primes in (x/2,x] equals n.

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Author

Keywords

Examples

References

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Crossrefs

Programs

Mathematica

Formula

Extensions

A164371 The smaller members of twin prime pairs which are non-Ramanujan primes.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Perl

Formula

Extensions

A174602 Smallest prime that begins a run of n Ramanujan primes that are consecutive primes.

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Author

A182365 The largest prime in some interval of the form (2prime(k),2prime(k+1)) if this interval contains at least 2 primes.