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%I #39 Feb 08 2015 08:49:22
%S 7,23,47,67,71,103,127,167,179,191,223,227,263,307,359,367,431,463,
%T 479,487,491,547,571,587,599,631,643,647,719,739,787,811,823,839,887,
%U 907,1019,1031,1051,1063,1087,1151,1223,1279,1303,1319,1399
%N Ramanujan primes R_(4,3)(n): a(n) is the smallest number such that if x >= a(n), then pi_(4,3)(x) - pi_(4,3)(x/2) >= n, where pi_(4,3)(x) is the number of primes==3 (mod 4) <= x.
%C All terms are primes==3 (mod 4).
%C A general conception of generalized Ramanujan numbers, see in Section 6 of the Shevelev, Greathouse IV, & Moses link.
%C We conjecture that for all n >= 1, a(n) <= A104272(3*n). This conjecture is based on observation that, if interval (x/2, x] contains >= 3*n primes, then at least n of them are of the form 4*k+3.
%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 lim(a(n)/prime(4*n)) = 1 as n tends to infinity.
%t Table[1+NestWhile[#1-1&,A104272[[3 k]],Count[Mod[Select[Range@@{Floor[#1/2+1],#1},PrimeQ],4],3]>=k&],{k,1,10}] using the code nn=1000;A104272=Table[0,{nn}];s=0;Do[If[PrimeQ[k],s++];If[PrimeQ[k/2],s--];If[s<nn,A104272[[s+1]]=k],{k,Prime[3*nn]}];A104272=A104272+1 (* _T. D. Noe_, Nov 15 2010 *)
%Y Cf. A104272, A185004, A185005, A185006.
%K nonn
%O 1,1
%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Dec 18 2012
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