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
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