1,1

From Vladimir Shevelev, Dec 08 2016: (Start)

Define P = exp(gamma)*log(log(k)), where gamma is Euler's constant A001620. The sequence lists numbers k for which phi(k) < k/P, where phi(k) is Euler's function A000010.

In 1909, Landau proved that for each eps>0, there exist infinitely many k for which phi(k) < k/P', where P' = exp(gamma-eps)*log(log(k)). In 1983 Nicolas strengthened Landau's result showing that there exist infinitely many k for which phi(k) < k/P. So this sequence is infinite.

All terms are even, except for 3,5,9 and 15. See proof in [Choie et al., Theorem 2.1]. (End)

E. Landau, Handbuch der Lehre yon der Verteilung der Primzahlen, 2 vols., Leipzig, Teubner, 1909. Reprinted in 1953 by Chelsea Publishing Co., New York.

Peter J. C. Moses, Table of n, a(n) for n = 1..5000 (first 2357 terms from T. D. Noe)

Y. Choie, N. Lichiardopol, P. Moree and P. Sole, On Robin's criterion for the Riemann hypothesis, J. Theor. Nombr. Bord. 19 (2) (2007), 357-372.

J.-L. Nicolas, Petites valeurs de la fonction d'Euler, J. Number Theory 17, no.3 (1983), 375-388.

Eric Weisstein's World of Mathematics, Totient Function.

Superset of A227243.

Cf. A000010 (phi), A001620 (gamma), A279161.

Sequence in context: A001957 A184484 A253897 * A358973 A063977 A371663

Adjacent sequences: A100963 A100964 A100965 * A100967 A100968 A100969

Eric W. Weisstein, Nov 23 2004

Edited by N. J. A. Sloane, Jan 04 2017

approved