A307382 Number of septic primitive Dirichlet characters modulo n.

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, 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, 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

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

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).

Cf. A319101 (number of solutions to x^7 == 1 (mod n)).

Sequence in context: A067483 A186685 A341882 * A339629 A331428 A323487

Adjacent sequences: A307379 A307380 A307381 * A307383 A307384 A307385

Keyword

nonn,mult

Author

Jianing Song, Apr 06 2019

Extensions

More terms from Antti Karttunen, Aug 22 2019

Status

approved