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%I #40 Jan 09 2024 08:49:26
%S 0,0,0,0,0,0,0,0,0,6,0,5,0,2,4,2,0,4,2,3,0,0,2,3,6,0,4,0,2,2,2,0,0,3,
%T 0,2,0,7,0,2,3,16,0,2,0,2,2,3,0,3,2,2,5,2,2,8,3,0,2,0,2,2,0,7,2,4,4,0,
%U 3,0
%N a(n) is the least r > 1 for which the interval (r*(2*n-1), r*(2*n+1)) contains no prime, or 0 if no such r exists.
%C In the first 50000 terms, the largest value is a(7333) = 37.
%C It is clear that a(1)=0, since it follows from the Bertrand postulate, which states that, for k>1, between k and 2*k there is a prime. This statement was proved first by P. Chebyshev and later by S. Ramanujan.
%C The equations a(2)=a(3)=a(4)=0 could be proved with the uniform positions, using Theorem 30 for generalized Ramanujan numbers from the Shevelev link. For proof of the equations a(n)=0 for n=5,...,9,11,13,17, etc., n<16597 we used a known result of L. Schoenfeld (1976) which states that for m>2010760, between m and m*(1+1/16597) there is always a prime, and, for 16597 <= n < 28314000, a stronger result of O. Ramaré and Y. Saouter (2003) which states that, for m >= 10726905041, between m*(1-1/28314000) and m there is always a prime.
%H Peter J. C. Moses, <a href="/A218850/b218850.txt">Table of n, a(n) for n = 1..20000</a>
%H J. Nagura, <a href="http://projecteuclid.org/euclid.pja/1195570997">On the interval containing at least one prime number</a>, Proc. Japan Acad., 28 (1952), 177-181.
%H S. Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram24.html">A proof of Bertrand's postulate</a>, J. Indian Math. Soc., 11 (1919), 181-182.
%H O. Ramaré and Y. Saouter, <a href="http://dx.doi.org/10.1016/S0022-314X(02)00029-X">Short effective intervals containing primes</a>, J. Number Theory, 98(2003), 10-33.
%H L. Schoenfeld, <a href="http://dx.doi.org/10.1090/S0025-5718-1976-0457374-X">Sharper bounds for the Chebyshev functions theta(x) and psi(x). II</a>, Math. Comp. 30 (1975) 337-360.
%H Vladimir 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 Vladimir Shevelev, Charles R. Greathouse IV, and 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> [math.NT], 2012.
%Y Cf. A218831, A218769.
%K nonn
%O 1,10
%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Nov 07 2012