A089684 Numbers k such that 2*phi(k) > k.

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127

Offset: 1

Author

Benoit Cloitre, Jan 16 2004

Keywords

• nonn
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Amiram Eldar, Plot of a(n)/n for n = 2^(20..34).
• 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.
• Vaclav Kotesovec, Plot of a(n)/n for n = 1..1000000.

Crossrefs

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

Cf. A036798, the missing odd numbers.

Complement of A119432.

Programs

Mathematica

lst={}; Do[If[2*EulerPhi[n]>n, AppendTo[lst, n]], {n, 200}]; lst (* T. D. Noe *)
Select[ Range[130], 2EulerPhi[ # ] > # &] (* Robert G. Wilson v, Jan 16 2004 *)

PARI

is(k) = 2*eulerphi(k) > k; \\ Amiram Eldar, Dec 01 2024

Formula

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

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