OFFSET
1,29
COMMENTS
a(n) is the number of primitive Dirichlet characters modulo n such that all entries are 0 or a seventh-power root of unity.
Mobius transform of A319101. Every term is 0 or a power of 6.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
FORMULA
Multiplicative with a(p^e) = 6 if p^e = 49 or p == 1 (mod 7) and e = 1, otherwise 0.
EXAMPLE
Let w = exp(2*Pi/7). For n = 29, the 6 septic primitive Dirichlet characters modulo n are:
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];
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];
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];
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];
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];
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],
so a(29) = 6.
MATHEMATICA
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 *)
PROG
(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))
(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
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Jianing Song, Apr 06 2019
EXTENSIONS
More terms from Antti Karttunen, Aug 22 2019
STATUS
approved