

A066500


Numbers k such that 5 divides phi(k).


9



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
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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, yx^k = y^5 = 1, yxy^(1) = x^r> is a nonabelian 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



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



KEYWORD

nonn


AUTHOR



EXTENSIONS

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


STATUS

approved



