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A104272
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Ramanujan primes R_n: 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.
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148
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2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127, 149, 151, 167, 179, 181, 227, 229, 233, 239, 241, 263, 269, 281, 307, 311, 347, 349, 367, 373, 401, 409, 419, 431, 433, 439, 461, 487, 491, 503, 569, 571, 587, 593, 599, 601, 607, 641, 643, 647, 653, 659
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OFFSET
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1,1
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COMMENTS
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Referring to his proof of Bertrand's postulate, Ramanujan states a generalization: "From this we easily deduce that pi(x) - pi(x/2) >= 1, 2, 3, 4, 5, ..., if x >= 2, 11, 17, 29, 41, ..., respectively." Since the a(n) are prime (by their minimality), I call them "Ramanujan primes."
See the additional references and links mentioned in A143227.
2n log 2n < a(n) < 4n log 4n for n >= 1, and prime(2n) < a(n) < prime(4n) if n > 1. Also, a(n) ~ prime(2n) as n -> infinity.
Shanta Laishram has proved that a(n) < prime(3n) for all n >= 1.
a(n) - 3n log 3n is sometimes positive, but negative with increasing frequency as n grows since a(n) ~ 2n log 2n. There should be a constant m such that for n >= m we have a(n) < 3n log 3n.
A good approximation to a(n) = R_n for n in [1..1000] is A162996(n) = round(k*n * (log(k*n)+1)), with k = 2.216 determined empirically from the first 1000 Ramanujan primes, which approximates the {k*n}-th prime number which in turn approximates the n-th Ramanujan prime and where abs(A162996(n) - R_n) < 2 * sqrt(A162996(n)) for n in [1..1000]. Since R_n ~ prime(2n) ~ 2n * (log(2n)+1) ~ 2n * log(2n), while A162996(n) ~ prime(k*n) ~ k*n * (log(k*n)+1) ~ k*n * log(k*n), 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). - Daniel Forgues, Jul 29 2009
Let p_n be the n-th prime. If p_n >= 3 is in the sequence, then all integers (p_n+1)/2, (p_n+3)/2, ..., (p_(n+1)-1)/2 are composite numbers. - Vladimir Shevelev, Aug 12 2009
Denote by q(n) the prime which is the nearest from the right to a(n)/2. Then there exists a prime between a(n) and 2q(n). Converse, generally speaking, is not true, i.e., there exist primes outside the sequence, but possess such property (e.g., 109). - Vladimir Shevelev, Aug 14 2009
The Mathematica program FasterRamanujanPrimeList uses Laishram's result that a(n) < prime(3n).
About 46% of primes < 19000 are Ramanujan primes. About 78% of the lesser of twin primes < 19000 are Ramanujan primes.
About 15% of primes < 19000 are the lesser of twin primes. About 26% of Ramanujan primes < 19000 are the lesser of twin primes.
A reason for the jumps is in Section 7 of "Ramanujan primes and Bertrand's postulate" and in Section 4 of "Ramanujan Primes: Bounds, Runs, Twins, and Gaps". (See the arXiv link for a corrected version of Table 1.)
See Shapiro 2008 for an exposition of Ramanujan's proof of his generalization of Bertrand's postulate. (End)
The (10^n)-th R prime: 2, 97, 1439, 19403, 242057, 2916539, 34072993, 389433437, .... - Robert G. Wilson v, May 07 2011, updated Aug 02 2012
The number of R primes < 10^n: 1, 10, 72, 559, 4459, 36960, 316066, 2760321, .... - Robert G. Wilson v, Aug 02 2012
a(n) = R_n = R_{0.5,n} in "Generalized Ramanujan Primes."
Or the largest prime x such that the number of primes in (x/2,x] equals n. This equivalent definition underlines an important analogy between Ramanujan and Labos primes (cf. A080359). - Vladimir Shevelev, Apr 29 2012
The questions on R_n - prime(2n) in A233739 have been answered by Christian Axler in "On generalized Ramanujan primes". - Jonathan Sondow, Feb 13 2014
Srinivasan's Lemma (2014): prime(k-n) < prime(k)/2 if R_n = prime(k) and n > 1. Proof: By the minimality of R_n, the interval (prime(k)/2,prime(k)] contains exactly n primes and so prime(k-n) < prime(k)/2. - Jonathan Sondow, May 10 2014
For some n and k, we see that A168421(k) = a(n) so as to form a chain of primes similar to a Cunningham chain. For example (and the first example), A168421(2) = 7, links a(2) = 11 = A168421(3), links a(3) = 17 = A168421(4), links a(4) = 29 = A168421(6), links a(6) = 47. Note that the links do not have to be of a form like q = 2*p+1 or q = 2*p-1. - John W. Nicholson, Feb 22 2015
Extending Sondow's 2010 comments: About 48% of primes < 10^9 are Ramanujan primes. About 76% of the lesser of twin primes < 10^9 are Ramanujan primes. - Dana Jacobsen, Sep 06 2015
Sondow, Nicholson, and Noe's 2011 conjecture that pi(R_{m*n}) <= m*pi(R_n) for m >= 1 and n >= N_m (see A190413, A190414) was proved for n > 10^300 by Shichun Yang and Alain Togbé in 2015. - Jonathan Sondow, Dec 01 2015
Berliner, Dean, Hook, Marr, Mbirika, and McBee (2016) prove in Theorem 18 that the graph K_{m,n} is prime for n >= R_{m-1}-m; see A291465. - Jonathan Sondow, May 21 2017
Okhotin (2012) uses Ramanujan primes to prove Lemma 8 in "Unambiguous finite automata over a unary alphabet." - Jonathan Sondow, May 30 2017
Sepulcre and Vidal (2016) apply Ramanujan primes in Remark 9 of "On the non-isolation of the real projections of the zeros of exponential polynomials." - Jonathan Sondow, May 30 2017
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REFERENCES
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Srinivasa 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.
Harold N. Shapiro, Ramanujan's idea, Section 9.3B in Introduction to the Theory of Numbers, Dover, 2008.
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LINKS
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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.
Adam H. Berliner, N. Dean, J. Hook, A. Marr, A. Mbirika, C. McBee, Coprime and prime labelings of graphs, arXiv preprint arXiv:1604.07698 [math.CO], 2016; Journal of Integer Sequences, Vol. 19 (2016), #16.5.8.
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FORMULA
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a(n) = 1 + max{k: pi(k) - pi(k/2) = n - 1}.
a(n) = max{prime p: pi(p) - pi(p/2) = n} (see Shevelev 2012). - Jonathan Sondow, Mar 23 2016
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EXAMPLE
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a(1) = 2 is Bertrand's postulate: pi(x) - pi(x/2) >= 1 for all x >= 2.
a(2) = 11 because a(2) < 8 log 8 < 17 and pi(n) - pi(n/2) > 1 for n = 16, 15, ..., 11 but pi(10) - pi(5) = 1.
Consider a(9)=71. Then the nearest prime > 71/2 is 37, and between a(9) and 2*37, that is, between 71 and 74, there exists a prime (73). - Vladimir Shevelev, Aug 14 2009 [corrected by Jonathan Sondow, Jun 17 2013]
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MAPLE
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local R;
if n = 1 then
return 2;
end if;
R := ithprime(3*n-1) ; # upper limit Laishram's thrm Thrm 3 arXiv:1105.2249
while true do
if A056171(R) = n then # Defn. 1. of Shevelev JIS 14 (2012) 12.1.1
return R ;
end if;
R := prevprime(R) ;
end do:
end proc:
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MATHEMATICA
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(RamanujanPrimeList[n_] := With[{T=Table[{k, PrimePi[k]-PrimePi[k/2]}, {k, Ceiling[N[4*n*Log[4*n]]]}]}, Table[1+First[Last[Select[T, Last[ # ]==i-1&]]], {i, 1, n}]]; RamanujanPrimeList[54]) (* Jonathan Sondow, Aug 15 2009 *)
(FasterRamanujanPrimeList[n_] := With[{T=Table[{k, PrimePi[k]-PrimePi[k/2]}, {k, Prime[3*n]}]}, Table[1+First[Last[Select[T, Last[ # ]==i-1&]]], {i, 1, n}]]; FasterRamanujanPrimeList[54])
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]}]; R=R+1 (* T. D. Noe, Nov 15 2010 *)
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PROG
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(Perl) use ntheory ":all"; my $r = ramanujan_primes(1000); say "[@$r]"; # Dana Jacobsen, Sep 06 2015
(PARI) ramanujan_prime_list(n) = {my(L=vector(n), s=0, k=1); for(k=1, prime(3*n)-1, if(isprime(k), s++); if(k%2==0 && isprime(k/2), s--); if(s<n, L[s+1] = k+1)); L} \\ Satish Bysany, Mar 02 2017
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CROSSREFS
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Cf. A000720, A007053, A014085, A060715, A084139, A084140, A143223, A143224, A143225, A143226, A143227, A080360, A080359, A164368, A164288, A164554, A164333, A164294, A164371, A190303.
Cf. A162996 (Round(kn * (log(kn)+1)), with k = 2.216 as an approximation of R_n = n-th Ramanujan Prime).
Cf. A163160 (Round(kn * (log(kn)+1)) - R_n, where k = 2.216 and R_n = n-th Ramanujan prime).
Cf. A178127 (Lesser of twin Ramanujan primes), A178128 (Lesser of twin primes if it is a Ramanujan prime).
Cf. A181671 (number of Ramanujan primes less than 10^n).
Cf. A190124 (constant of summation: 1/a(n)^2).
Cf. A190413, A190414, A212493, A212541, A233739, A233822, A277718, A277719, A164952, A290394, A291465.
Not to be confused with the Ramanujan numbers or Ramanujan tau function, A000594.
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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