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A307382 Number of septic primitive Dirichlet characters modulo n.

%I A307382 #20 Feb 19 2021 03:37:57
%S A307382 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,
%T A307382 0,0,0,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A307382 0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0
%N A307382 Number of septic primitive Dirichlet characters modulo n.
%C A307382 a(n) is the number of primitive Dirichlet characters modulo n such that all entries are 0 or a seventh-power root of unity.
%C A307382 Mobius transform of A319101. Every term is 0 or a power of 6.
%H A307382 Antti Karttunen, <a href="/A307382/b307382.txt">Table of n, a(n) for n = 1..65537</a>
%F A307382 Multiplicative with a(p^e) = 6 if p^e = 49 or p == 1 (mod 7) and e = 1, otherwise 0.
%e A307382 Let w = exp(2*Pi/7). For n = 29, the 6 septic primitive Dirichlet characters modulo n are:
%e A307382   Chi_1 = [0, 1, w, w^5, w^2, w, w^6, w^5, w^3, w^3, w^2, w^4, 1, w^4, w^6, w^6, w^4, 1, w^4, w^2, w^3, w^3, w^5, w^6, w, w^2, w^5, w, 1];
%e A307382   Chi_2 = [0, 1, w^2, w^3, w^4, w^2, w^5, w^3, w^6, w^6, w^4, w, 1, w, w^5, w^5, w, 1, w, w^4, w^6, w^6, w^3, w^5, w^2, w^4, w^3, w^2, 1];
%e A307382   Chi_3 = [0, 1, w^3, w, w^6, w^3, w^4, w, w^2, w^2, w^6, w^5, 1, w^5, w^4, w^4, w^5, 1, w^5, w^6, w^2, w^2, w, w^4, w^3, w^6, w, w^3, 1];
%e A307382   Chi_4 = [0, 1, w^4, w^6, w, w^4, w^3, w^6, w^5, w^5, w, w^2, 1, w^2, w^3, w^3, w^2, 1, w^2, w, w^5, w^5, w^6, w^3, w^4, w, w^6, w^4, 1];
%e A307382   Chi_5 = [0, 1, w^5, w^4, w^3, w^5, w^2, w^4, w, w, w^3, w^6, 1, w^6, w^2, w^2, w^6, 1, w^6, w^3, w, w, w^4, w^2, w^5, w^3, w^4, w^5, 1];
%e A307382   Chi_6 = [0, 1, w^6, w^2, w^5, w^6, w, w^2, w^4, w^4, w^5, w^3, 1, w^3, w, w, w^3, 1, w^3, w^5, w^4, w^4, w^2, w, w^6, w^5, w^2, w^6, 1],
%e A307382 so a(29) = 6.
%t A307382 f[7, 2] = 6; f[p_, e_] := If[Mod[p, 7] == 1 && e == 1, 6, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 16 2020 *)
%o A307382 (PARI) a(n)=sum(d=1, n, if(n%d==0, moebius(n/d)*sum(i=1, d, if((i^7-1)%d, 0, 1)), 0))
%o A307382 (PARI) A307382(n) = { my(f=factor(n)); prod(i=1, #f~, if(((7==f[i,1])&&(2==f[i,2]))||((1==(f[i,1]%7))&&(1==f[i,2])),6,0)); }; \\ _Antti Karttunen_, Aug 22 2019
%Y A307382 Number of k-th power primitive Dirichlet characters modulo n: A114643 (k=2), A160498 (k=3), A160499 (k=4), A307380 (k=5), A307381 (k=6), this sequence (k=7), A329272 (k=8).
%Y A307382 Cf. A319101 (number of solutions to x^7 == 1 (mod n)).
%K A307382 nonn,mult
%O A307382 1,29
%A A307382 _Jianing Song_, Apr 06 2019
%E A307382 More terms from _Antti Karttunen_, Aug 22 2019