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Numbers k such that 2*phi(k) <= k.
4

%I #17 Oct 15 2020 03:58:58

%S 2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,

%T 50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,

%U 96,98,100,102,104,105,106,108,110,112,114,116,118,120,122,124,126,128,130

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

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

%C 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

%C 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

%H Amiram Eldar, <a href="/A119432/b119432.txt">Table of n, a(n) for n = 1..10000</a>

%H Mitsuo Kobayashi, <a href="https://doi.org/10.1142/S1793042116500445">A generalization of a series for the density of abundant numbers</a>, International Journal of Number Theory, Vol. 12, No. 3 (2016), pp. 671-677.

%H Charles R. Wall, <a href="https://doi.org/10.1090/S0025-5718-1972-0327701-9">Density bounds for Euler's function</a>, Mathematics of Computation, Vol. 26, No. 119 (1972), pp. 779-783.

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

%t Select[Range[130], 2*EulerPhi[#] <= # &] (* _Amiram Eldar_, Feb 29 2020 *)

%o (PARI) is(n)=2*eulerphi(n)<=n \\ _Charles R Greathouse IV_, Oct 26 2015

%Y Disjoint union of A119434 and A299174. - _Amiram Eldar_, Oct 15 2020

%Y Cf. A000010, A054741, A119433.

%K nonn

%O 1,1

%A _Franklin T. Adams-Watters_, May 19 2006