login
A088232
Numbers k such that 3 does not divide phi(k).
9
1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 15, 16, 17, 20, 22, 23, 24, 25, 29, 30, 32, 33, 34, 40, 41, 44, 46, 47, 48, 50, 51, 53, 55, 58, 59, 60, 64, 66, 68, 69, 71, 75, 80, 82, 83, 85, 87, 88, 89, 92, 94, 96, 100, 101, 102, 106, 107, 110, 113, 115, 116, 118, 120, 121, 123, 125, 128
OFFSET
1,2
COMMENTS
n such that the congruence x^3 == 1 mod(n) has only the trivial solution x=1 i.e. A060839(n) = 1 . Complement of sequence A066498.
Let U(n) be the group of positive integers coprime to n under mod n multiplication. Let U(n)^3 = {x^3: x is an element of U(n)}. These are the n such that U(n) = U(n)^3. - Geoffrey Critzer, Jun 07 2015
In other words, numbers divisible neither by 9 nor by any primes of the form 6k+1. - Ivan Neretin, Sep 03 2015
The asymptotic density of this sequence is 0 (Dressler, 1975). - Amiram Eldar, Jul 23 2020
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Robert E. Dressler, A property of the phi and sigma_j functions, Compositio Mathematica, Vol. 31, No. 2 (1975), pp. 115-118.
FORMULA
a(n) ~ k n sqrt(log(n)) for some constant k. k appears to be around 1.08. [Charles R Greathouse IV, Feb 14 2012]
MAPLE
select(t -> numtheory:-phi(t) mod 3 <> 0, [$1..1000]); # Robert Israel, Sep 04 2015
MATHEMATICA
Prepend[Position[Table[Union[Select[Range[n], CoprimeQ[#, n] &]] ==
Union[Mod[Select[Range[n], CoprimeQ[#, n] &]^3, n]], {n, 1, 155}], True], 1] // Flatten (* Geoffrey Critzer, Jun 07 2015 *)
Select[Range[140], !Divisible[EulerPhi[#], 3]&] (* Harvey P. Dale, Sep 23 2017 *)
PROG
(PARI) is(n)=eulerphi(n)%3 \\ Charles R Greathouse IV, Feb 04 2013
CROSSREFS
Cf. A000010, A066498 (complement).
Positions of 1's in A060839, of 0's in A354099, of nonzeros in A074942.
Cf. also A329963.
Sequence in context: A180636 A182625 A333909 * A070994 A291686 A057197
KEYWORD
nonn
AUTHOR
Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 03 2003
EXTENSIONS
More terms from Ray Chandler, Nov 05 2003
STATUS
approved