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 A066498 Numbers k such that 3 divides phi(k). 16
 7, 9, 13, 14, 18, 19, 21, 26, 27, 28, 31, 35, 36, 37, 38, 39, 42, 43, 45, 49, 52, 54, 56, 57, 61, 62, 63, 65, 67, 70, 72, 73, 74, 76, 77, 78, 79, 81, 84, 86, 90, 91, 93, 95, 97, 98, 99, 103, 104, 105, 108, 109, 111, 112, 114, 117, 119, 122, 124, 126, 127, 129, 130, 133 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Numbers k such that x^3 == 1 (mod k) has solutions 1 < x < k. Terms are multiple of 9 or of a prime of the form 6k+1. If k is a term of this sequence, then G = 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 The asymptotic density of this sequence is 1 (Dressler, 1975). - Amiram Eldar, Mar 21 2021 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. EXAMPLE 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. MATHEMATICA Select[Range, Divisible[EulerPhi[#], 3]&] (* Harvey P. Dale, Jan 12 2011 *) PROG (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 CROSSREFS Complement of A088232. Cf. A000010, A066499, A066500, A066501, A066502. A007645 gives the primes congruent to 1 mod 3. Column k=2 of A277915. Sequence in context: A056528 A055565 A196088 * A102306 A066962 A067020 Adjacent sequences: A066495 A066496 A066497 * A066499 A066500 A066501 KEYWORD nonn AUTHOR Benoit Cloitre, Jan 04 2002 EXTENSIONS Simpler definition from Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 25 2003 Corrected and extended by Ray Chandler, Nov 05 2003 STATUS approved

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Last modified December 2 14:00 EST 2023. Contains 367524 sequences. (Running on oeis4.)