The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A054741 Numbers m such that totient(m) < cototient(m). 11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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
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.
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]
EXAMPLE
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.
MATHEMATICA
Select[ Range[300], 2EulerPhi[ # ] < # &] (* Robert G. Wilson v, Jan 10 2004 *)
PROG
(PARI) is(n)=n>2*eulerphi(n) \\ Charles R Greathouse IV, Sep 13 2013
CROSSREFS
A177712 is a subsequence. Complement: A115405.
Positions of negative terms in A083254.
Cf. A323170 (characteristic function).
Complement of A000079\{1} within A119432.
Sequence in context: A056868 A069209 A060702 * A177712 A176552 A176587
KEYWORD
nonn
AUTHOR
Labos Elemer, Apr 26 2000
EXTENSIONS
Erroneous comment removed by Antti Karttunen, Jan 05 2019
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 29 12:08 EDT 2024. Contains 372940 sequences. (Running on oeis4.)