

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
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OFFSET

1,1


COMMENTS

For powers of 2, the two function values are equal.
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/(p1) as k > oo; given that p/(p1) = p^k/phi(p^k) for k >= 1.  Peter Munn, Nov 24 2020


LINKS



FORMULA

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


MATHEMATICA



PROG



CROSSREFS

Positions of negative terms in A083254.
Cf. A323170 (characteristic function).


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



