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A220474 Chebyshev numbers C_v(n) for v=10/9: a(n) is the smallest number such that if x>=a(n), then theta(x)-theta(9*x/10)>=n*log(x), where theta(x)=sum_{prime p<=x}log p.

%I A220474 #39 Dec 04 2016 19:46:30
%S A220474 223,227,269,349,359,569,587,593,739,809,857,991,1009,1019,1259,1481,
%T A220474 1483,1487,1489,1861,1867,1993,1997,2003,2027,2267,2269,2657,2671,
%U A220474 2687,2689,2699,3181,3187,3307,3313,3319,3323,3457,3461,3491,3527,3529,3581,3623,3769,4049,4201,4391,4481
%N A220474 Chebyshev numbers C_v(n) for v=10/9: a(n) is the smallest number such that if x>=a(n), then theta(x)-theta(9*x/10)>=n*log(x), where theta(x)=sum_{prime p<=x}log p.
%C A220474 All terms are primes.
%C A220474 Up to a(99)=9029, all terms are (10/9)-Ramanujan numbers as in Shevelev's link; up to 9029, the only missing (10/9)-Ramanujan number is 127.
%H A220474 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 A220474 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 A220474 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 A220474 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 A220474 a(n)<=prime(31*(n+1)).
%t A220474 k=9; xs=Table[{m,Ceiling[x/.FindRoot[(x (-1300+Log[x]^4))/Log[x]^5==(k+1) m,{x,f[(k+1) m]-1},AccuracyGoal->Infinity,PrecisionGoal->20,WorkingPrecision->100]]},{m,1,101}]; Table[{m,1+NestWhile[#-1&,xs[[m]][[2]],(1/Log[#1]Plus@@Log[Select[Range[Floor[(k #1)/(k+1)]+1,#1],PrimeQ]]&)[#]>m&]},{m,1,100}] (* _Peter J. C. Moses_, Dec 20 2012 *)
%t A220474 (* Assuming range of x is from a(n) to 2*a(n) *) Clear[a, theta]; theta[x_] := theta[x] = Sum[Log[p], {p, Table[Prime[k], {k, 1, PrimePi[x]}]}] // N; a[0] = 211(* just to speed-up computation *); a[n_] := a[ n] = (t = Table[an = Prime[pi]; Table[{an, x >= an && theta[x] - theta[9*x/10] >= n*Log[x]}, {x, an, 2*an}], {pi, PrimePi[a[n-1]], 31*(n+1)}]; 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, 50}] (* _Jean-François Alcover_, Feb 11 2013 *)
%Y A220474 Cf. A220293, A220462, A220463.
%K A220474 nonn
%O A220474 1,1
%A A220474 _Vladimir Shevelev_, _Charles R Greathouse IV_ and _Peter J. C. Moses_, Dec 15 2012
%E A220474 More terms from _Jean-François Alcover_, Feb 11 2013