A088232 Numbers k such that 3 does not divide phi(k).

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.

Kevin Ford, Florian Luca, and Pieter Moree, Values of the Euler phi-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields, arXiv:1108.3805 [math.NT], 2011-2012.

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

Adjacent sequences: A088229 A088230 A088231 * A088233 A088234 A088235

Keyword

nonn

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 03 2003

Extensions

More terms from Ray Chandler, Nov 05 2003

Status

approved