A091886 - OEIS (original) (raw)

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

COMMENTS

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.

MATHEMATICA

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