[seqfan] Bound for A233739(n) / n
John W. Nicholson
reddwarf2956 at yahoo.com
Sun Nov 6 03:44:21 CET 2016
In A233739, Jonathan Sondow asked "Is a(n)/n bounded?" and Christian Axler has proved it is bounded.
Using plot2 to graph a(n)/n:
https://oeis.org/plot2a?name1=A233739&name2=A27&tform1=absolute+value&tform2=untransformed&shift=0&radiop1=ratio&drawlines=true
It looks like this bounded value is between 1 and 2. To prove this I tried to use the fact that:
1) For all R(k) > 3251 or for all k > 201, R(k) < 2k log R(k) . (1)
2) p_(2k) < R(k). (2)
3) p_n > n*(log(p_n)-1.1). (3)
But I get values of 0 < a(n)/ n < 2.2.
Can anyone do better?
(1) From A214934 and C. Axler, <a href="http://arxiv.org/abs/1401.7179">On generalized Ramanujan primes</a>, arXiv preprint arXiv:1401.7179, 2014.
(2) From J. Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009-2010.
(3) From PIERRE DUSART, ESTIMATES OF SOME FUNCTIONS OVER PRIMES WITHOUT R.H. 2010, (6.6) with x = p_n > 60184.
John W. Nicholson
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