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

1,1

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.

Formula

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

Mathematica

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

Prog

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

Crossrefs

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

Cf. A000010, A054741, A119433.

Sequence in context: A055966 A087113 A366846 * A005843 A004275 A317108

Adjacent sequences: A119429 A119430 A119431 * A119433 A119434 A119435

Keyword

nonn

Author

Franklin T. Adams-Watters, May 19 2006

Status

approved