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A160499 Number of quartic primitive Dirichlet characters modulo n. 7
1, 0, 1, 1, 3, 0, 1, 2, 0, 0, 1, 1, 3, 0, 3, 4, 3, 0, 1, 3, 1, 0, 1, 2, 0, 0, 0, 1, 3, 0, 1, 0, 1, 0, 3, 0, 3, 0, 3, 6, 3, 0, 1, 1, 0, 0, 1, 4, 0, 0, 3, 3, 3, 0, 3, 2, 1, 0, 1, 3, 3, 0, 0, 0, 9, 0, 1, 3, 1, 0, 1, 0, 3, 0, 0, 1, 1, 0, 1, 12, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Also called biquadratic primitive Dirichlet characters.

Primitive Dirichlet characters of both order 2 & order 4 are included.

a(n) is the number of primitive Dirichlet characters modulo n such that all entries are 0 or a fourth-power root of unity (1, i, -1 and -i). - Jianing Song, Feb 27 2019

Mobius transform of A073103. - Jianing Song, Mar 02 2019

LINKS

Jianing Song, Table of n, a(n) for n = 1..10000

Steven R. Finch, Cubic and quartic characters [Broken link]

Steven R. Finch, Cubic and quartic characters

FORMULA

Multiplicative with a(4) = 1, a(8) = 2, a(16) = 4, a(2^e) = 0 for e = 1 or e >= 5; for odd primes p, a(p) = 3 if p == 1 (mod 4) and 1 if p == 3 (mod 4), a(p^e) = 0 for e >= 2. - Jianing Song, Mar 02 2019

EXAMPLE

From Jianing Song, Mar 02 2019: (Start)

For n = 5, the 3 quartic primitive Dirichlet characters modulo n are [0, 1, -1, -1, 1], [0, 1, i, -i, -1] and [0, 1, -i, i, -1], so a(5) = 3.

For n = 16, the 4 quartic primitive Dirichlet characters modulo n are [0, 1, 0, i, 0, i, 0, 1, 0, -1, 0, -i, 0, -i, 0, -1], [0, 1, 0, -i, 0, -i, 0, 1, 0, -1, 0, i, 0, i, 0, -1], [0, 1, 0, i, 0, -i, 0, -1, 0, -1, 0, -i, 0, i, 0, 1] and [0, 1, 0, -i, 0, i, 0, -1, 0, -1, 0, i, 0, -i, 0, 1], so a(16) = 4. (End)

MATHEMATICA

f[n_] := Sum[If[Mod[k^4 - 1, n] == 0, 1, 0], {k, 1, n}]; a[n_] := Sum[ MoebiusMu[n/d]*f[d], {d, Divisors[n]}]; Table[a[n], {n, 2, 81}] (* Jean-Fran├žois Alcover, Jun 19 2013 *)

PROG

(PARI) a(n)=sum(d=1, n, if(n%d==0, moebius(n/d)*sum(i=1, d, if((i^4-1)%d, 0, 1)), 0)) \\ Steven Finch, Jun 09 2009

CROSSREFS

Cf. A114643 (number of quadratic primitive Dirichlet characters modulo n), A160498 (number of cubic primitive Dirichlet characters modulo n).

Cf. A073103 (number of solutions to x^4 == 1 (mod n)).

Sequence in context: A126308 A094923 A303301 * A329272 A274876 A065718

Adjacent sequences:  A160496 A160497 A160498 * A160500 A160501 A160502

KEYWORD

mult,nonn

AUTHOR

Steven Finch, May 15 2009

EXTENSIONS

a(1) = 1 prepended by Jianing Song, Feb 27 2019

STATUS

approved

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Last modified November 20 12:13 EST 2019. Contains 329335 sequences. (Running on oeis4.)