

A066498


Numbers k such that 3 divides phi(k).


13



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
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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 = <x, yx^k = y^3 = 1, yxy^(1) = x^r> is a nonabelian 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


LINKS

Harry J. Smith, Table of n, a(n) for n=1..1000


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[150], 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

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



