A054741 Numbers m such that totient(m) < cototient(m).

6, 10, 12, 14, 18, 20, 22, 24, 26, 28, 30, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 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, 130, 132, 134, 136

Offset: 1

Author

Labos Elemer, Apr 26 2000

Keywords

• nonn

Comments

For powers of 2, the two function values are equal.

Numbers m such that m/phi(m) > 2. - Charles R Greathouse IV, Sep 13 2013

Numbers m such that A173557(m)/A007947(m) < 1/2. - Antti Karttunen, Jan 05 2019

Numbers m such that there are powers of m that are abundant. This follows from abundancy and totient being multiplicative, with the abundancy for prime p of p^k being asymptotically p/(p-1) as k -> oo; given that p/(p-1) = p^k/phi(p^k) for k >= 1. - Peter Munn, Nov 24 2020

Examples

			For m = 20, phi(20) = 8, cototient(20) = 20 - phi(20) = 12, 8 = phi(20) < 20-phi(20) = 12; for m = 21, the opposite holds: phi = 12, 21-phi = 8.
		

Links

• Charles R Greathouse IV, 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

A177712 is a subsequence. Complement: A115405.

Positions of negative terms in A083254.

Cf. A000010, A007947, A051953, A005408, A036798, A089684, A173557.

Cf. A323170 (characteristic function).

Complement of A000079\{1} within A119432.

Programs

Mathematica

Select[ Range[300], 2EulerPhi[ # ] < # &] (* Robert G. Wilson v, Jan 10 2004 *)

PARI

is(n)=n>2*eulerphi(n) \\ Charles R Greathouse IV, Sep 13 2013

Formula

m such that A000010(m) < A051953(m).

a(n) seems to be asymptotic to c*n with c=1.9566...... - Benoit Cloitre, Oct 20 2002 [It is an old theorem that a(n) ~ cn for some c, for any sequence of the form "m/phi(m) > k". - Charles R Greathouse IV, May 28 2015] [c is in the interval (1.9540, 1.9562) (Kobayashi, 2016). - Amiram Eldar, Feb 14 2021]

Extensions

Erroneous comment removed by Antti Karttunen, Jan 05 2019