A066498 - OEIS (original) (raw)

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

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

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

CROSSREFS

A007645 gives the primes congruent to 1 mod 3.

EXTENSIONS

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

Corrected and extended by Ray Chandler, Nov 05 2003