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

1,1

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 = <x, y|x^k = y^3 = 1, yxy^(-1) = x^r> 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

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.

Example

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.

Mathematica

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

Prog

(PARI) { n=0; for (m=1, 10^10, if (eulerphi(m)%3 == 0, write("b066498.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 18 2010

Crossrefs

Complement of A088232.

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

A007645 gives the primes congruent to 1 mod 3.

Column k=2 of A277915.

Sequence in context: A056528 A055565 A196088 * A102306 A066962 A067020

Adjacent sequences: A066495 A066496 A066497 * A066499 A066500 A066501

Keyword

nonn

Author

Benoit Cloitre, Jan 04 2002

Extensions

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

Corrected and extended by Ray Chandler, Nov 05 2003

Status

approved