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

Author

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

Keywords

• nonn

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.

Crossrefs

Cf. A000010, A066498 (complement).

Positions of 1's in A060839, of 0's in A354099, of nonzeros in A074942.

Cf. also A329963.

Programs

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 *)

PARI

is(n)=eulerphi(n)%3 \\ Charles R Greathouse IV, Feb 04 2013

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]

Extensions

More terms from Ray Chandler, Nov 05 2003