

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
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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
The asymptotic density of this sequence is 1 (Dressler, 1975).  Amiram Eldar, Mar 21 2021


LINKS



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

A007645 gives the primes congruent to 1 mod 3.


KEYWORD

nonn


AUTHOR



EXTENSIONS

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


STATUS

approved



