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A185006 Ramanujan primes R_(4,1)(n): a(n) is the smallest number such that if x >= a(n), then pi_(4,1)(x) - pi_(4,1)(x/2) >= n, where pi_(4,1)(x) is the number of primes==1 (mod 4) <= x. 3

%I

%S 13,37,41,89,97,109,149,229,233,241,257,277,281,317,349,397,401,409,

%T 421,433,569,593,601,641,653,661,709,757,761,821,929,937,941,953,977,

%U 997,1009,1021,1049,1061,1093,1097,1117,1193,1213,1237,1249

%N Ramanujan primes R_(4,1)(n): a(n) is the smallest number such that if x >= a(n), then pi_(4,1)(x) - pi_(4,1)(x/2) >= n, where pi_(4,1)(x) is the number of primes==1 (mod 4) <= x.

%C All terms are primes==1 (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+1.

%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], 1] >= 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, A185005, A185004, A185007.

%K nonn

%O 1,1

%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Dec 18 2012

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Last modified June 16 04:18 EDT 2021. Contains 345055 sequences. (Running on oeis4.)