A084140 - OEIS

%I #70 Jul 10 2021 20:34:17

%S 2,6,9,15,21,24,30,34,36,49,51,54,64,75,76,84,90,91,114,115,117,120,

%T 121,132,135,141,154,156,174,175,184,187,201,205,210,216,217,220,231,

%U 244,246,252,285,286,294,297,300,301,304,321,322,324,327,330,339,360,364

%N a(n) is the smallest number j such that if x >= j there are at least n primes between x and 2x exclusively.

%C For all m >= a(n) there are at least n primes between m and 2m exclusively. This calculation relies on the fact that pi(2m) - pi(m) > m/(3*log(m)) for m >= 5. This is one more than the terms of A084139 with offset changed from 0 to 1.

%C For n > 5889, pi(2n) - pi(n) > f(2, 2n) - f(3, n) where f(k, x) = x/log x * (1 + 1/log x + k/(log x)^2). This may be useful for checking larger terms. The constant 3 can be improved at the cost of an increase in the constant 5889. - _Charles R Greathouse IV_, May 02 2012

%C A168421(n) = nextprime(a(n)), where nextprime(x) is the next prime >= x. - _John W. Nicholson_, Dec 21 2012

%C a(1) = ceiling((A104272(1)+1)/2) modifies the only even prime, 2; which has been stated, in Formula, as a(1) = A104272(1); for all others, a(n) = (A104272(n)+1)/2 = ceiling ((A104272(n)+1)/2). - _John W. Nicholson_, Dec 24 2012

%C Srinivasan's Lemma (2014): previousprime(a(n)) = p_(k-n) < (p_k)/2, where the n-th Ramanujan Prime R_n is the k-th prime p_k, and with n > 1. Proof: By the minimality of R_n, the interval ((p_k)/2,p_k] contains exactly n primes, so p_(k-n) < (p_k)/2. - Copied and adapted from a comment by _Jonathan Sondow_ in A168421 by _John W. Nicholson_, Feb 17 2015

%D Paulo Ribenboim, The Little Book of Big Primes, Springer-Verlag, 1991, p. 140.

%D Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag, 2004, p. 181.

%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 version</a>, arXiv:1212.2785 [math.NT], 2012.

%H Anitha Srinivasan, <a href="http://www.emis.de/journals/INTEGERS/papers/o19/o19.Abstract.html">An upper bound for Ramanujan primes</a>, Integers, 19 (2014), #A19.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BertrandsPostulate.html">Bertrand's Postulate</a>.

%F a(1) = A104272(1); for n >= 2, a(n) = (A104272(n)+1)/2. - _Vladimir Shevelev_, Dec 07 2012

%F a(n) = ceiling((A104272(n)+1)/2) for n >= 1. - _John W. Nicholson_, Dec 24 2012

%e a(11)=51 since there are at least 11 primes between m and 2m for all m >= 51 and this is not true for any m < 51. Although a(100)=720 is not listed, for all m >= 720, there are at least 100 primes between m and 2m.

%t nn = 60;

%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[3 nn]}];

%t A104272 = R + 1;

%t Ceiling[(A104272 + 1)/2] (* _Jean-François Alcover_, Nov 07 2018, after _T. D. Noe_ in A104272 *)

%Y Cf. A108954, A060715, A060756, A084138, A084139, A084141, A084142, A168421, A168425.

%K nonn

%O 1,1

%A _Harry J. Smith_, May 15 2003