

A119432


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


4



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
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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. 671677.
Charles R. Wall, Density bounds for Euler's function, Mathematics of Computation, Vol. 26, No. 119 (1972), pp. 779783.


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 A004275 * A005843 A317108 A317440
Adjacent sequences: A119429 A119430 A119431 * A119433 A119434 A119435


KEYWORD

nonn


AUTHOR

Franklin T. AdamsWatters, May 19 2006


STATUS

approved



