A119432 Numbers k such that 2*phi(k) <= k.

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 105, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130

Offset: 1

Author

Franklin T. Adams-Watters, May 19 2006

Keywords

• nonn

Comments

Equivalently, numbers k such that totient(k) <= cototient(k).

Using the primes up to 23 it is possible to show that this sequence has (lower) density greater than 0.51. - Charles R Greathouse IV, Oct 26 2015

The asymptotic density of this sequence is in the interval (0.51120, 0.51176) (Kobayashi, 2016, improving the bounds 0.5105 and 0.5241 that were given by Wall, 1972). - Amiram Eldar, Oct 15 2020

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Mitsuo Kobayashi, A generalization of a series for the density of abundant numbers, International Journal of Number Theory, Vol. 12, No. 3 (2016), pp. 671-677.
• Charles R. Wall, Density bounds for Euler's function, Mathematics of Computation, Vol. 26, No. 119 (1972), pp. 779-783.

Crossrefs

Disjoint union of A119434 and A299174. - Amiram Eldar, Oct 15 2020

Cf. A000010, A054741, A119433.

Programs

Mathematica

Select[Range[130], 2*EulerPhi[#] <= # &] (* Amiram Eldar, Feb 29 2020 *)

PARI

is(n)=2*eulerphi(n)<=n \\ Charles R Greathouse IV, Oct 26 2015

Formula

Elements of A054741 together with all 2^n for n>0.