%I #32 Nov 14 2014 06:56:09
%S 13,37,41,67,73,97,127,137,173,179,181,211,229,239,263,307,311,347,
%T 367,379,431,433,443,449,479,487,541,563,587,599,607,641,643,673,739,
%U 757,787,797,809,823,827,859,937,967,997,1019,1031,1039,1049,1061,1087
%N Chebyshev numbers C_v(n) for v=3/2: a(n) is the smallest number such that if x>=a(n), then theta(x)-theta(2*x/3)>=n*log(x), where theta(x) = sum_{prime p<=x} log p.
%C All terms are primes.
%C Up to a(97)=2333, only four terms of the sequence (a(33)=643, a(34)=673, a(76)=1721 and a(77)=1741) are not (3/2)-Ramanujan numbers as in Shevelev's link; up to 2333, the only (3/2)-Ramanujan numbers missing from the sequence are 2, 617, 653, 709, 1709, 1733, and 1747.
%H N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, <a href="http://arxiv.org/abs/1108.0475">Generalized Ramanujan primes</a>, arXiv 2011.
%H N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, <a href="http://link.springer.com/chapter/10.1007/978-1-4939-1601-6_1">Generalized Ramanujan primes</a>, Combinatorial and Additive Number Theory, Springer Proc. in Math. & Stat., CANT 2011 and 2012, Vol. 101 (2014), 1-13
%H V. Shevelev, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Shevelev/shevelev19.html">Ramanujan and Labos primes, their generalizations, and classifications of primes</a>, J. Integer Seq. 15 (2012) Article 12.5.4
%H Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Moses/moses1.html">On intervals (kn, (k+1)n) containing a prime for all n>1</a>, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. <a href="http://arxiv.org/abs/1212.2785">arXiv:1212.2785</a>
%F a(n)<=prime(4*(n+1)).
%t (* Assuming range of x is from a(n) to 2*a(n) *) theta[x_] := Sum[Log[p], {p, Table[Prime[k], {k, 1, PrimePi[x]}]}]; Clear[a]; a[0] = 2; a[n_] := a[n] = (t = Table[{an, x >= an && theta[x] - theta[2*(x/3)] >= n*Log[x]}, {an, a[n - 1], Prime[4*(n + 1)]}, {x, an, 2*an}]; sp = t // Flatten[#, 1] & // Sort // Split[#, #1[[1]] == #2[[1]] &] &; Select[sp, And @@ (#[[All, 2]]) &] // First // First // First); Table[Print[a[n]]; a[n], {n, 1, 51}] (* _Jean-François Alcover_, Jan 24 2013 *)
%Y Cf. A220293.
%K nonn
%O 1,1
%A _Vladimir Shevelev_, _Charles R Greathouse IV_ and _Peter J. C. Moses_, Dec 15 2012