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

Offset: 1

Author

Benoit Cloitre, Jan 04 2002

Keywords

• nonn

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 = 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

Examples

			x^5 == 1 (mod 11) has solutions 1 < x < 11, namely {3,4,5,9}.
		

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.

Crossrefs

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

Column k=3 of A277915.

Programs

Mathematica

Select[Range[250], Divisible[EulerPhi[#], 5] &] (* Amiram Eldar, May 23 2022 *)

PARI

isok(k) = { eulerphi(k)%5 == 0 } \\ Harry J. Smith, Feb 18 2010

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).

Extensions

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

Extended by Ray Chandler, Nov 06 2003