A066498 Numbers k such that 3 divides phi(k).

7, 9, 13, 14, 18, 19, 21, 26, 27, 28, 31, 35, 36, 37, 38, 39, 42, 43, 45, 49, 52, 54, 56, 57, 61, 62, 63, 65, 67, 70, 72, 73, 74, 76, 77, 78, 79, 81, 84, 86, 90, 91, 93, 95, 97, 98, 99, 103, 104, 105, 108, 109, 111, 112, 114, 117, 119, 122, 124, 126, 127, 129, 130, 133

Offset: 1

Author

Benoit Cloitre, Jan 04 2002

Keywords

• nonn

Comments

Numbers k such that x^3 == 1 (mod k) has solutions 1 < x < k.

Terms are multiple of 9 or of a prime of the form 6k+1.

If k is a term of this sequence, then G = is a non-abelian group of order 3k, where 1 < r < n and r^3 == 1 (mod k). For example, G can be the subgroup of GL(2, Z_k) generated by x = {{1, 1}, {0, 1}} and y = {{r, 0}, {0, 1}}. - Jianing Song, Sep 17 2019

The asymptotic density of this sequence is 1 (Dressler, 1975). - Amiram Eldar, Mar 21 2021

Examples

			If n < 7 then x^3 = 1 (mod n) has no solution 1 < x < n, but {2,4} are solutions to x^3 == 1 (mod 7), hence a(1) = 7.
		

Links

• Harry J. Smith, Table of n, a(n) for n=1..1000
• Robert E. Dressler, A property of the phi and sigma_j functions, Compositio Mathematica, Vol. 31, No. 2 (1975), pp. 115-118.

Crossrefs

Complement of A088232.

Cf. A000010, A066499, A066500, A066501, A066502.

A007645 gives the primes congruent to 1 mod 3.

Column k=2 of A277915.

Programs

Mathematica

Select[Range[150], Divisible[EulerPhi[#], 3]&] (* Harvey P. Dale, Jan 12 2011 *)

PARI

isok(k)={ eulerphi(k)%3 == 0 } \\ Harry J. Smith, Feb 18 2010

Extensions

Simpler definition from Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 25 2003

Corrected and extended by Ray Chandler, Nov 05 2003