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A066498 Numbers k such that 3 divides phi(k).

%I A066498 #49 Dec 11 2024 11:34:09
%S A066498 7,9,13,14,18,19,21,26,27,28,31,35,36,37,38,39,42,43,45,49,52,54,56,
%T A066498 57,61,62,63,65,67,70,72,73,74,76,77,78,79,81,84,86,90,91,93,95,97,98,
%U A066498 99,103,104,105,108,109,111,112,114,117,119,122,124,126,127,129,130,133
%N A066498 Numbers k such that 3 divides phi(k).
%C A066498 Numbers k such that x^3 == 1 (mod k) has solutions 1 < x < k.
%C A066498 Terms are multiple of 9 or of a prime of the form 6k+1.
%C A066498 If k is a term of this sequence, then G = <x, y|x^k = y^3 = 1, yxy^(-1) = x^r> is a non-abelian group of order 3k, where 1 < r < n and r^3 == 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 A066498 The asymptotic density of this sequence is 1 (Dressler, 1975). - _Amiram Eldar_, Mar 21 2021
%H A066498 Harry J. Smith, <a href="/A066498/b066498.txt">Table of n, a(n) for n=1..1000</a>
%H A066498 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.
%e A066498 If n < 7 then x^3 = 1 (mod n) has no solution 1 < x < n, but {2,4} are solutions to x^3 == 1 (mod 7), hence a(1) = 7.
%t A066498 Select[Range[150], Divisible[EulerPhi[#], 3]&] (* _Harvey P. Dale_, Jan 12 2011 *)
%o A066498 (PARI) isok(k)={ eulerphi(k)%3 == 0 } \\ _Harry J. Smith_, Feb 18 2010
%Y A066498 Complement of A088232.
%Y A066498 Cf. A000010, A066499, A066500, A066501, A066502.
%Y A066498 A007645 gives the primes congruent to 1 mod 3.
%Y A066498 Column k=2 of A277915.
%K A066498 nonn
%O A066498 1,1
%A A066498 _Benoit Cloitre_, Jan 04 2002
%E A066498 Simpler definition from Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 25 2003
%E A066498 Corrected and extended by _Ray Chandler_, Nov 05 2003