

A066502


Numbers k such that 7 divides phi(k).


10



29, 43, 49, 58, 71, 86, 87, 98, 113, 116, 127, 129, 142, 145, 147, 172, 174, 196, 197, 203, 211, 213, 215, 226, 232, 239, 245, 254, 258, 261, 281, 284, 290, 294, 301, 319, 337, 339, 343, 344, 348, 355, 377, 379, 381, 387, 392, 394, 406, 421, 422, 426, 430
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OFFSET

1,1


COMMENTS

Related to the equation x^7 == 1 (mod k): sequence gives values of k such there are solutions 1 < x < k of x^7 == 1 (mod k).
If k is a term of this sequence, then G = <x, yx^k = y^7 = 1, yxy^(1) = x^r> is a nonabelian group of order 7k, where 1 < r < n and r^7 == 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, May 23 2022


LINKS



FORMULA

a(n) are the numbers generated by 7^2 = 49 and all primes congruent to 1 mod 7 (A045465). Hence sequence gives all k such that k == 0 (mod A045465(n)) for some n > 1 or k == 0 (mod 49).


EXAMPLE

x^7 == 1 (mod k) has solutions 1 < x < k for k = 29, 43, 49, ...


MATHEMATICA

Select[Range[500], Divisible[EulerPhi[#], 7]&] (* Harvey P. Dale, Apr 12 2012 *)


PROG

(PARI) { n=0; for (m=1, 10^10, if (eulerphi(m)%7 == 0, write("b066502.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 18 2010


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS

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


STATUS

approved



