cp's OEIS Frontend

A066500 Numbers k such that 5 divides phi(k).

%I A066500 #27 Dec 11 2024 11:33:54
%S A066500 11,22,25,31,33,41,44,50,55,61,62,66,71,75,77,82,88,93,99,100,101,110,
%T A066500 121,122,123,124,125,131,132,142,143,150,151,154,155,164,165,175,176,
%U A066500 181,183,186,187,191,198,200,202,205,209,211,213,217,220,225,231,241
%N A066500 Numbers k such that 5 divides phi(k).
%C A066500 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).
%C A066500 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
%C A066500 The asymptotic density of this sequence is 1 (Dressler, 1975). - _Amiram Eldar_, May 23 2022
%H A066500 Harry J. Smith, <a href="/A066500/b066500.txt">Table of n, a(n) for n = 1..1000</a>
%H A066500 Robert E. Dressler, <a href="http://www.numdam.org/item/?id=CM_1975__31_2_115_0">A property of the phi and sigma_j functions</a>, Compositio Mathematica, Vol. 31, No. 2 (1975), pp. 115-118.
%F A066500 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).
%e A066500 x^5 == 1 (mod 11) has solutions 1 < x < 11, namely {3,4,5,9}.
%t A066500 Select[Range[250], Divisible[EulerPhi[#], 5] &] (* _Amiram Eldar_, May 23 2022 *)
%o A066500 (PARI) isok(k) = { eulerphi(k)%5 == 0 } \\ _Harry J. Smith_, Feb 18 2010
%Y A066500 Cf. A000010, A045453, A066498, A066499, A066501, A066502.
%Y A066500 Column k=3 of A277915.
%K A066500 nonn
%O A066500 1,1
%A A066500 _Benoit Cloitre_, Jan 04 2002
%E A066500 Simpler definition from Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 25 2003
%E A066500 Extended by _Ray Chandler_, Nov 06 2003