|
%I #27 Oct 28 2018 09:14:45
%S 2,71,101,181,239,241,269,349,373,409,419,433,439,491,593,599,601,607,
%T 647,653,659,823,827,857,947,1021,1031,1061,1063,1091,1103,1301,1427,
%U 1429,1447,1451,1489,1553,1559,1567,1601,1607,1609,1789,1867,1871,1913,1999,2003
%N Ramanujan primes A104272(n) for which A104272(n) = A080359(n).
%C For every n>=1, A104272(n) >= A080359(n), and the sequence shows where the inequality becomes an equality.
%C Let prime(m) < a(n)/2 < prime(m+1); then there exist primes p<q such that p is in the interval (2*Prime(m), a(n)) and q is in the interval (a(n), 2*Prime(m+1)).
%C For example, a(2) = 71, 31 < a(2)/2 < 37 and intervals (62,71), (71,74) contain the primes p = 67 and q = 73 respectively.
%C Let us call a prime p compatible with another prime q, if the intervals (p/2,q/2) and (p,q], if q>p, (or intervals (q/2,p/2) and (q,p], if q<p) contain the same number of primes. If p is compatible with no other prime, we call it a peculiar prime. The sequence lists the peculiar primes. [_Vladimir Shevelev_, Apr 25 2012]
%H V. Shevelev, <a href="http://arXiv.org/abs/0908.2319">On critical small intervals containing primes</a>, arXiv:0908.2319 [math.NT] [From _Vladimir Shevelev_, Aug 20 2009]
%H 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 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>, J. Integer Seq. 14 (2011) Article 11.6.2
%F All solutions of the equation A104272(x)=A080359(x) are x=pi(a(n))-pi(a(n)/2). - Vladimir Shevelev, Apr 25 2012
%e a(2)=71, such that 31<71/2<37, and we see that p=67 is in interval (62, 71) and q=73 is in interval (71, 74).
%t nn = 200; t = Table[0, {nn+1}]; s = 0;
%t Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s <= nn && t[[s+1]] == 0, t[[s+1]] = k], {k, Prime[3nn]}
%t ];
%t A080359 = Rest[t];
%t R = Table[0, {nn}]; s = 0;
%t Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s < nn, R[[s+1]] = k], {k, Prime[3nn]}
%t ];
%t A104272 = R+1;
%t Intersection[A104272, A080359] (* _Jean-François Alcover_, Oct 28 2018, after _T. D. Noe_ in A104272 *)
%Y Cf. A104272, A080359, A164368, A164333, A164288, A164294, A212493, A212541.
%K nonn
%O 1,1
%A _Vladimir Shevelev_, Aug 15 2009
%E Terms beyond 659 from _R. J. Mathar_, Dec 17 2009
|
