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%I #70 Aug 06 2017 03:19:56
%S 0,0,0,2,0,4,2,3,0,2,3,2,2,0,6,2,2,3,2,6,3,2,4,2,2,7,2,2,4,3,2,2,4,2,
%T 4,4,2,2,3,2,3,2,2,3,2,2,2,3,2,4,3,2,3,4,2,3,2,2,2,2,2,2,4,2,5,2,2,3,
%U 3,2,2,2,2,4,4,2,2,3,2,2,3,2,4,2,2,3,2
%N a(n) is the least r > 1 for which the interval (r*n, r*(n+1)) contains no prime, or a(n)=0 if no such r exists.
%C In the first 50000000 terms a(n) is 0 only for n=1, 2, 3, 5, 9, 14. In the same range the largest value of a(n) is 16 at n=2540, 77384, 1679690, 3240054, 13078899.
%C a(1)=0 is "Bertrand's postulate," which states that there is always a prime between k and 2*k. This was first proved by P. Chebyshev.
%C Note that the equations a(2) = a(3) = 0 are results of M. El. Buchraoui and A. Loo respectively and could be proved with the uniform positions, using Theorem 30 for generalized Ramanujan numbers from the Shevelev link. The equation a(5) = 0 follows from the result of J. Nagura. For proof of the equations a(9)=a(14)=0, we used a known result of L. Schoenfeld (1976) that states that for n>2010760, between n and n*(1+1/16597) there is always a prime.
%H Peter J. C. Moses., <a href="/A218831/b218831.txt">Table of n, a(n) for n = 1..20000</a>
%H N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, <a href="http://arxiv.org/abs/1108.0475">Generalized Ramanujan primes</a>, arXiv 2011.
%H M. El Bachraoui, <a href="http://www.m-hikari.com/ijcms-password/ijcms-password13-16-2006/elbachraouiIJCMS13-16-2006.pdf">Primes in the interval [2n,3n]</a>, Int. J. Contemp. Math. Sciences 1:13 (2006), pp. 617-621.
%H N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, <a href="https://doi.org/10.1007/978-1-4939-1601-6_1">Generalized Ramanujan primes</a>, Combinatorial and Additive Number Theory, Springer Proc. in Math. & Stat., CANT 2011 and 2012, Vol. 101 (2014), 1-13
%H A. Loo, <a href="http://www.m-hikari.com/ijcms-2011/37-40-2011/looIJCMS37-40-2011.pdf">On the primes in the interval [3n,4n]</a>, International Journal of Contemporary Mathematical Sciences, volume 6, number 38, pages 1871-1882, 2011.
%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 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 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>
%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.
%F a(n) = 0 <=> A220315(k) = n for some k. - _Jonathan Sondow_, Aug 04 2017
%t rmax = 100; a[n_] := Catch[ For[r = 2, r <= rmax, r++, If[PrimePi[r*n] == PrimePi[r*(n + 1)], Throw[r], If[r == rmax, Throw[0]]]]]; Table[ a[n] , {n, 1, 87}] (* _Jean-François Alcover_, Dec 13 2012 *)
%Y Cf. A218769, A220268, A220269, A220273, A220274, A220281, A220315.
%K nonn
%O 1,4
%A _Vladimir Shevelev_, _Charles R Greathouse IV_ and _Peter J. C. Moses_, Nov 07 2012
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