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A066498 Numbers k such that 3 divides phi(k). 16

%I #45 Mar 21 2021 12:10:29

%S 7,9,13,14,18,19,21,26,27,28,31,35,36,37,38,39,42,43,45,49,52,54,56,

%T 57,61,62,63,65,67,70,72,73,74,76,77,78,79,81,84,86,90,91,93,95,97,98,

%U 99,103,104,105,108,109,111,112,114,117,119,122,124,126,127,129,130,133

%N Numbers k such that 3 divides phi(k).

%C Numbers k such that x^3 == 1 (mod k) has solutions 1 < x < k.

%C Terms are multiple of 9 or of a prime of the form 6k+1.

%C 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 The asymptotic density of this sequence is 1 (Dressler, 1975). - _Amiram Eldar_, Mar 21 2021

%H Harry J. Smith, <a href="/A066498/b066498.txt">Table of n, a(n) for n=1..1000</a>

%H 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 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 Select[Range[150], Divisible[EulerPhi[#], 3]&] (* _Harvey P. Dale_, Jan 12 2011 *)

%o (PARI) { n=0; for (m=1, 10^10, if (eulerphi(m)%3 == 0, write("b066498.txt", n++, " ", m); if (n==1000, return)) ) } \\ _Harry J. Smith_, Feb 18 2010

%Y Complement of A088232.

%Y Cf. A000010, A066499, A066500, A066501, A066502.

%Y A007645 gives the primes congruent to 1 mod 3.

%Y Column k=2 of A277915.

%K nonn

%O 1,1

%A _Benoit Cloitre_, Jan 04 2002

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

%E Corrected and extended by _Ray Chandler_, Nov 05 2003

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Last modified March 28 16:12 EDT 2024. Contains 371254 sequences. (Running on oeis4.)