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A066502 Numbers k such that 7 divides phi(k).

%I A066502 #28 Dec 11 2024 11:33:50
%S A066502 29,43,49,58,71,86,87,98,113,116,127,129,142,145,147,172,174,196,197,
%T A066502 203,211,213,215,226,232,239,245,254,258,261,281,284,290,294,301,319,
%U A066502 337,339,343,344,348,355,377,379,381,387,392,394,406,421,422,426,430
%N A066502 Numbers k such that 7 divides phi(k).
%C A066502 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).
%C A066502 If k is a term of this sequence, then G = <x, y|x^k = y^7 = 1, yxy^(-1) = x^r> is a non-abelian 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
%C A066502 The asymptotic density of this sequence is 1 (Dressler, 1975). - _Amiram Eldar_, May 23 2022
%H A066502 Harry J. Smith, <a href="/A066502/b066502.txt">Table of n, a(n) for n = 1..1000</a>
%H A066502 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.
%F A066502 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).
%e A066502 x^7 == 1 (mod k) has solutions 1 < x < k for k = 29, 43, 49, ...
%t A066502 Select[Range[500],Divisible[EulerPhi[#],7]&] (* _Harvey P. Dale_, Apr 12 2012 *)
%o A066502 (PARI) isok(k) = { eulerphi(k)%7 == 0 } \\ _Harry J. Smith_, Feb 18 2010
%Y A066502 Cf. A045465, A066498, A066499, A066500, A066501, A000010.
%Y A066502 Column k=4 of A277915.
%K A066502 nonn
%O A066502 1,1
%A A066502 _Benoit Cloitre_, Jan 04 2002
%E A066502 Simpler definition from Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 25 2003