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A193507 Ramanujan primes of the second kind: a(n) is the smallest prime such that if prime x >= a(n), then pi(x) - pi(x/2) >= n, where pi(x) is the number of primes <= x.

%I A193507 #63 Feb 16 2025 08:33:15
%S A193507 2,3,13,19,31,43,53,61,71,73,101,103,109,131,151,157,173,181,191,229,
%T A193507 233,239,241,251,269,271,283,311,313,349,353,373,379,409,419,421,433,
%U A193507 439,443,463,491,499,509,571,577,593,599,601,607,613,643,647,653,659
%N A193507 Ramanujan primes of the second kind: a(n) is the smallest prime such that if prime x >= a(n), then pi(x) - pi(x/2) >= n, where pi(x) is the number of primes <= x.
%C A193507 Apparently A168425 and the 2. - R. J. Mathar, Aug 25 2011
%C A193507 An odd prime p is in the sequence iff the previous prime is Ramanujan. The Ramanujan primes and the Ramanujan primes of the second kind are the mutually wrapping up sequences: a(1)<=R_1<=a(2)<=R_2<=a(3)<=R_3<=.... . - _Vladimir Shevelev_, Aug 29 2011
%C A193507 All terms of the sequence are in A194598. - _Vladimir Shevelev_, Aug 30 2011
%H A193507 V. Shevelev, <a href="http://arxiv.org/abs/0909.0715">Ramanujan and Labos primes, their generalizations and classifications of primes</a>, arXiv:0909.0715 [math.NT], 2009-2011.
%H A193507 J. Sondow, <a href="https://mathworld.wolfram.com/RamanujanPrime.html">Ramanujan Prime in MathWorld</a>
%H A193507 J. Sondow, J. W. Nicholson, and T. D. Noe, <a href="http://arxiv.org/abs/1105.2249"> Ramanujan Primes: Bounds, Runs, Twins, and Gaps</a>, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2.
%F A193507 A080359(n) <= a(n) <= A104272(n) = R_n (Cf. A194184, A194186).
%F A193507 a(n)>p_(2*n-1); a(n)~p_{2n} (Cf. properties of R_n in A104272 and the above comment). - _Vladimir Shevelev_, Aug 28 2011
%e A193507 Since R_2=11 (see A104272), then for x >= 11, we have pi(x) - pi(x/2) >= 2. However, if to consider only prime x, then we see that, for x=7,5,3, pi(x) - pi(x/2)= 2, but pi(2) - pi(1)= 1. Therefore, already for prime x>=3, we have pi(x) - pi(x/2) >= 2. Thus a(2)=3.
%t A193507 nn = 120; (* nn=120 returns 54 terms *)
%t A193507 R = Table[0, {nn}]; s = 0;
%t A193507 Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s < nn, R[[s + 1]] = k], {k, Prime[3 nn]}];
%t A193507 A104272 = R + 1;
%t A193507 Join[{2}, Select[Prime[Range[nn]], MemberQ[A104272, NextPrime[#, -1]]&]] (* _Jean-François Alcover_, Nov 07 2018, after _T. D. Noe_ in A104272 *)
%Y A193507 Cf. A104272 (Ramanujan primes).
%K A193507 nonn
%O A193507 1,1
%A A193507 _Vladimir Shevelev_, Aug 18 2011