OFFSET
1,1
COMMENTS
Related to the equation x^5 == 1 (mod k): sequence gives values of k such there are solutions 1 < x < k of x^5 == 1 (mod k).
If k is a term of this sequence, then G = <x, y|x^k = y^5 = 1, yxy^(-1) = x^r> is a non-abelian group of order 5k, where 1 < r < n and r^5 == 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, May 23 2022
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.
FORMULA
EXAMPLE
x^5 == 1 (mod 11) has solutions 1 < x < 11, namely {3,4,5,9}.
MATHEMATICA
Select[Range[250], Divisible[EulerPhi[#], 5] &] (* Amiram Eldar, May 23 2022 *)
PROG
(PARI) { n=0; for (m=1, 10^10, if (eulerphi(m)%5 == 0, write("b066500.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
Extended by Ray Chandler, Nov 06 2003
STATUS
approved