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.
Steven R. Finch, Cubic and quartic characters.
Steven R. Finch, Quartic and Octic Characters Modulo n, arXiv:0907.4894 [math.NT], 2016.
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
Sum_{k=1..n} a(k) ~ c * n * log(n), where c = (7/(16*Pi*K^2)) * Product_{primes p == 1 (mod 4)} (1 - (5*p-3)/(p^2*(p+1))) = 0.1908767211685284480112237..., and K is the Landau-Ramanujan constant (A064533). - Amiram Eldar, Sep 16 2020
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-François Alcover, Jun 19 2013 *)
f[2, e_] := Which[e == 1, 0, e == 2, 1, e == 3, 2, e == 4, 4, e >= 5, 0]; f[p_, 1] := If[Mod[p, 4] == 1, 3, 1]; f[p_, e_] := 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^4-1)%d, 0, 1)), 0)) \\ Steven Finch, Jun 09 2009
CROSSREFS
KEYWORD
mult,nonn
AUTHOR
Steven Finch, May 15 2009
EXTENSIONS
a(1) = 1 prepended by Jianing Song, Feb 27 2019
STATUS
approved