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Integers k for which the open interval (k*m, (k+1)*m) contains a prime for all m > 1.
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%I #36 Feb 11 2021 06:18:20

%S 1,2,3,5,9,14

%N Integers k for which the open interval (k*m, (k+1)*m) contains a prime for all m > 1.

%C Shevelev, Greathouse, and Moses (2012) prove that if more terms exist, they are >= 5*10^7.

%H N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and J. Sondow, <a href="http://arxiv.org/abs/1108.0475">Generalized Ramanujan primes</a>, arXiv:1108.0475 [math.NT], 2011.

%H N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and 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 Vladimir Shevelev, Charles R. Greathouse IV, and Peter J. C. Moses, <a href="http://arxiv.org/abs/1212.2785">On intervals (kn,(k+1)n) containing a prime for all n>1</a>, arXiv:1212.2785 [math.NT], 2012.

%H J. Sondow, <a href="http://arxiv.org/abs/0907.5232">Ramanujan primes and Bertrand's postulate</a>, arXiv:0907.5232 [math.NT], 2009-2010; Amer. Math. Monthly, 116 (2009), 630-635.

%F a(n) = k for some n <=> A218831(k) = 0. - _Jonathan Sondow_, Aug 04 2017

%e a(1) = 1 because Bertrand's Postulate (proved by Chebyshev) implies that for any m > 1 there is a prime p with m < p < 2m.

%Y Cf. A104272, A218831.

%K nonn,more

%O 1,2

%A _Jonathan Sondow_, Dec 13 2012