%I #16 Apr 12 2022 04:24:06
%S 6,9,10,12,14,15,16,18,20,21,22,24,25,26,27,28,30,31,32,33,34,35,36,
%T 37,38,39,40,41,42,47,48,49,50,51,52,53,54,55,56,57,58,59,60,62,63,64,
%U 65,66,67,68,69,70,71,72,79,80,81,82,85,86,87,88,89,90,91,92,93,94,95
%N Values of k such that PrimePi(k)^2 < (e*k*PrimePi(k/e))/log(k), where e = 2.71828... (A001113).
%C From _Amiram Eldar_, Apr 12 2022: (Start)
%C Ramanujan proved that all sufficiently large values of k are in this sequence.
%C According to Berndt (1994), W. Galway found that the largest prime below 10^11 that is not in this sequence is 38358837677.
%C Hassani (2012) proved that assuming the Riemann hypothesis, all numbers >= 138766146692471228 are in this sequence.
%C Dudek and Platt (2015) proved that assuming the Riemann hypothesis 38358837682 is the largest number that is not in this sequence, and that unconditionally all numbers > exp(9658) are in this sequence.
%C Axler (2018) proved that the inequality holds for all numbers between 38358837683 and 10^19 and for all numbers > exp(9032).
%C Platt and Trudgian (2021) proved that the inequality holds for all numbers between 38358837683 and exp(58) and for all numbers > exp(3915).
%C Johnston (2021) proved that the inequality holds for all numbers between 38358837683 and exp(103).
%C Cully-Hugill and Johnston (2021) proved that the inequality holds for all numbers > exp(3604). (End)
%D Bruce C. Berndt, Ramanujan's Notebooks, Part IV, New York: Springer-Verlag, 1994, pp. 112-113.
%D S. Ramanujan, Notebooks, 2 vols., Tata Institute of Fundamental Research, Bombay, 1957, 2nd notebook, p. 310.
%H Amiram Eldar, <a href="/A091886/b091886.txt">Table of n, a(n) for n = 1..10000</a>
%H Christian Axler, <a href="http://math.colgate.edu/~integers/s61/s61.mail.html">Estimates for pi(x) for large values of x and Ramanujan's prime counting inequality</a>, Integers, Vol. 18 (2018), Article A61, 14pp.; <a href="https://arxiv.org/abs/1703.02407">arXiv preprint</a>, arXiv:1703.02407 [math.NT], 2017.
%H Michaela Cully-Hugill and Daniel R. Johnston, <a href="https://arxiv.org/abs/2111.10001">On the error term in the explicit formula of Riemann-von Mangoldt</a>, arXiv:2111.10001 [math.NT], 2021.
%H Adrian William Dudek, <a href="https://doi.org/10.25911/5d7638521bcc4">Explicit Estimates in the Theory of Prime Numbers</a>, Doctoral dissertation, The Australian National University, 2016; <a href="https://arxiv.org/abs/1611.07251">arXiv preprint</a>, arXiv:1611.07251 [math.NT], 2016,
%H Adrian W. Dudek and David J. Platt, <a href="https://doi.org/10.1080/10586458.2014.990118">On Solving a Curious Inequality of Ramanujan</a>, Experimental Mathematics, Vol. 24, No. 3 (2015), pp. 289-294; <a href="https://arxiv.org/abs/1407.1901">arXiv preprint</a>, arXiv:1407.1901 [math.NT], 2014.
%H Mehdi Hassani, <a href="https://doi.org/10.1007/s11139-011-9362-6">On an inequality of Ramanujan concerning the prime counting function</a>, The Ramanujan Journal, Vol. 28, No. 3 (2012), pp. 435-442; <a href="https://www.researchgate.net/profile/Mehdi-Hassani/publication/257642844_On_an_inequality_of_Ramanujan_concerning_the_prime_counting_function/links/5f61226b4585154dbbd53998">ResearchGate link</a>.
%H Daniel R. Johnston, <a href="https://arxiv.org/abs/2109.02249">Improving bounds on prime counting functions by partial verification of the Riemann hypothesis</a>, arXiv:2109.02249 [math.NT], 2021.
%H David Platt and Timothy Trudgian, <a href="https://doi.org/10.1090/mcom/3583">The error term in the prime number theorem</a>, Mathematics of Computation, Vol. 90, No. 328 (2021), pp. 871-881; <a href="https://arxiv.org/abs/1809.03134">arXiv preprint</a>, arXiv:1809.03134 [math.NT], 2018-2020.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeCountingFunction.html">Prime Counting Function</a>.
%t Select[Range[2, 100], PrimePi[#]^2 < (E*#*PrimePi[#/E])/Log[#] &] (* _Amiram Eldar_, Apr 12 2022 *)
%Y Cf. A000720, A001113.
%K nonn
%O 1,1
%A _Eric W. Weisstein_, Feb 08 2004
%E Offset corrected by _Amiram Eldar_, Apr 12 2022