This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A220269 #40 Nov 30 2019 01:28:35
%S A220269 2,8,11,17,26,38,40,41,48,57,68,68,70,87,96,100,108,109,110,115,136,
%T A220269 149,151,161,161,169,176,178,184,206,208,227,235,236,242,255,259,260,
%U A220269 263,272,297,299,305,320,356,358,359,371,371,372,378,386,389,392,400
%N A220269 a(n) is the smallest number, such that for all N >= a(n) there are at least n primes between 3*N and 4*N.
%H A220269 Peter J. C. Moses, <a href="/A220269/b220269.txt">Table of n, a(n) for n = 1..3000</a>
%H A220269 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 A220269 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 A220269 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 A220269 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 A220269 a(n) <= ceiling(R_(4/3)(n)/4), where R_v(n) (v>1) are generalized Ramanujan numbers (see Shevelev's link). In particular, for n >= 1, {R_(4/3)(n)}={11, 29, 59, 67, 101, 149, 157, 163, 191, 227, 269, 271, 307, 379, ...}. Moreover, if R_(4/3)(n) == 1 (mod 4), then a(n) = ceiling(R_(4/3)(n)/4).
%t A220269 nn = 60; t = Table[PrimePi[4 n] - PrimePi[3 n], {n, 10*nn}]; Join[{2}, Table[s = Flatten[Position[t, _?(# > n - 1 &)]]; i = Length[s]; While[i > 1 && s[[i]] - s[[i - 1]] == 1, i--]; s[[i]], {n, 2, nn}]] (* _Michael B. Porter_, after A220268 program by _T. D. Noe_, Feb 09 2013 *)
%Y A220269 Cf. A084140, A220268.
%K A220269 nonn
%O A220269 1,1
%A A220269 _Vladimir Shevelev_, _Charles R Greathouse IV_ and _Peter J. C. Moses_, Dec 09 2012