A066500 - OEIS

A066500

Numbers k such that 5 divides phi(k).

11, 22, 25, 31, 33, 41, 44, 50, 55, 61, 62, 66, 71, 75, 77, 82, 88, 93, 99, 100, 101, 110, 121, 122, 123, 124, 125, 131, 132, 142, 143, 150, 151, 154, 155, 164, 165, 175, 176, 181, 183, 186, 187, 191, 198, 200, 202, 205, 209, 211, 213, 217, 220, 225, 231, 241

(list; graph; refs; listen; history; text; internal format)

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

a(n) are the numbers generated by 5^2 = 25 and all primes congruent to 1 mod 5 (A045453). Hence sequence gives all k such that k == 0 (mod A045453(n)) for some n > 1 or k == 0 (mod 25).

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

Cf. A000010, A045453, A066498, A066499, A066501, A066502.

Column k=3 of A277915.

Sequence in context: A095779 A062055 A332516 * A234314 A258738 A160272

Adjacent sequences: A066497 A066498 A066499 * A066501 A066502 A066503

KEYWORD