A089684 - OEIS

%I #26 Dec 01 2024 04:09:54

%S 1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,

%T 49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,

%U 95,97,99,101,103,107,109,111,113,115,117,119,121,123,125,127

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

%H Amiram Eldar, <a href="/A089684/b089684.txt">Table of n, a(n) for n = 1..10000</a>

%H Amiram Eldar, <a href="/A089684/a089684_1.jpg">Plot of a(n)/n for n = 2^(20..34)</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 Vaclav Kotesovec, <a href="/A089684/a089684.jpg">Plot of a(n)/n for n = 1..1000000</a>.

%F Asymptotic to c*n with c = 2.045...

%F 2.04582 < c < 2.04818 (from the bounds on the asymptotic density of A119432 given by Kobayashi, 2016). - _Amiram Eldar_, Dec 01 2024

%t lst={}; Do[If[2*EulerPhi[n]>n, AppendTo[lst, n]], {n, 200}]; lst (* _T. D. Noe_ *)

%t Select[ Range[130], 2EulerPhi[ # ] > # &] (* _Robert G. Wilson v_, Jan 16 2004 *)

%o (PARI) is(k) = 2*eulerphi(k) > k; \\ _Amiram Eldar_, Dec 01 2024

%Y Cf. A000010, A036798, A067800 (nonprime n such that phi(n) > n/2).

%Y Cf. A036798, the missing odd numbers.

%Y Complement of A119432.

%K nonn,easy

%O 1,2

%A _Benoit Cloitre_, Jan 16 2004