A119434 Odd n such that 2*phi(n) < n.

105, 165, 195, 315, 495, 525, 585, 735, 825, 945, 975, 1155, 1365, 1485, 1575, 1755, 1785, 1815, 1995, 2145, 2205, 2415, 2475, 2535, 2625, 2805, 2835, 2925, 3003, 3045, 3135, 3255, 3315, 3465, 3675, 3705, 3795, 3885, 3927, 4095, 4125, 4305, 4389, 4455

Offset: 1

Author

Franklin T. Adams-Watters, May 19 2006

Keywords

• nonn

Comments

Obviously 2*phi(n) = n is impossible for odd n. Odd elements of A054741 and A119432. This is not the same as A036798. 684411 = 3*7*13*23*109 is in this sequence but not in A036798. (This is may not be the smallest such value.) The primitive elements of this sequence are A119433, excluding the initial 2

If n is in the sequence, then so is every odd multiple of n. - Robert Israel, Jan 06 2017

The asymptotic density of this sequence is in the interval (0.01120, 0.01176) (Kobayashi, 2016). It is 1/2 less than the asymptotic density of A119432. The number of terms below 10^k for k = 3, 4, ... are 11, 109, 1152, 11076, 111927, 1124091, 11224403, 112074112, ... - Amiram Eldar, Oct 15 2020

Links

• Robert Israel, 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.

Crossrefs

Cf. A000010, A036798, A054741, A118700, A119432, A119433, A299174.

Programs

Maple

select(t -> numtheory:-phi(t) < t/2, [seq(t,t=1..10000,2)]);

Mathematica

Select[Range[1, 10^4, 2], 2 EulerPhi[#] < #&] (* Jean-François Alcover, Apr 12 2019 *)

PARI

lista(nn) = forstep (n=1, nn, 2, if (n > 2*eulerphi(n), print1(n, ", "))) \\ Michel Marcus, Jul 04 2015

Formula

A036798 UNION A118700. - R. J. Mathar, Aug 08 2007

A119432 \ A299174. - Amiram Eldar, Oct 15 2020