A091886 Values of k such that PrimePi(k)^2 < (e*k*PrimePi(k/e))/log(k), where e = 2.71828... (A001113).

6, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 79, 80, 81, 82, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95

Offset

1,1

Comments

From Amiram Eldar, Apr 12 2022: (Start)

Ramanujan proved that all sufficiently large values of k are in this sequence.

According to Berndt (1994), W. Galway found that the largest prime below 10^11 that is not in this sequence is 38358837677.

Hassani (2012) proved that assuming the Riemann hypothesis, all numbers >= 138766146692471228 are in this sequence.

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.

Axler (2018) proved that the inequality holds for all numbers between 38358837683 and 10^19 and for all numbers > exp(9032).

Platt and Trudgian (2021) proved that the inequality holds for all numbers between 38358837683 and exp(58) and for all numbers > exp(3915).

Johnston (2021) proved that the inequality holds for all numbers between 38358837683 and exp(103).

Cully-Hugill and Johnston (2021) proved that the inequality holds for all numbers > exp(3604). (End)

References

Bruce C. Berndt, Ramanujan's Notebooks, Part IV, New York: Springer-Verlag, 1994, pp. 112-113.

S. Ramanujan, Notebooks, 2 vols., Tata Institute of Fundamental Research, Bombay, 1957, 2nd notebook, p. 310.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Christian Axler, Estimates for pi(x) for large values of x and Ramanujan's prime counting inequality, Integers, Vol. 18 (2018), Article A61, 14pp.; arXiv preprint, arXiv:1703.02407 [math.NT], 2017.

Michaela Cully-Hugill and Daniel R. Johnston, On the error term in the explicit formula of Riemann-von Mangoldt, arXiv:2111.10001 [math.NT], 2021.

Adrian William Dudek, Explicit Estimates in the Theory of Prime Numbers, Doctoral dissertation, The Australian National University, 2016; arXiv preprint, arXiv:1611.07251 [math.NT], 2016,

Adrian W. Dudek and David J. Platt, On Solving a Curious Inequality of Ramanujan, Experimental Mathematics, Vol. 24, No. 3 (2015), pp. 289-294; arXiv preprint, arXiv:1407.1901 [math.NT], 2014.

Mehdi Hassani, On an inequality of Ramanujan concerning the prime counting function, The Ramanujan Journal, Vol. 28, No. 3 (2012), pp. 435-442; ResearchGate link.

Daniel R. Johnston, Improving bounds on prime counting functions by partial verification of the Riemann hypothesis, arXiv:2109.02249 [math.NT], 2021.

David Platt and Timothy Trudgian, The error term in the prime number theorem, Mathematics of Computation, Vol. 90, No. 328 (2021), pp. 871-881; arXiv preprint, arXiv:1809.03134 [math.NT], 2018-2020.

Eric Weisstein's World of Mathematics, Prime Counting Function.

Mathematica

Select[Range[2, 100], PrimePi[#]^2 < (E*#*PrimePi[#/E])/Log[#] &] (* Amiram Eldar, Apr 12 2022 *)

Crossrefs

Cf. A000720, A001113.

Sequence in context: A103092 A337940 A104523 * A333357 A111774 A325170

Adjacent sequences: A091883 A091884 A091885 * A091887 A091888 A091889

Keyword

nonn

Author

Eric W. Weisstein, Feb 08 2004

Extensions

Offset corrected by Amiram Eldar, Apr 12 2022

Status

approved