OFFSET
1,7
COMMENTS
In the character table of the multiplicative group modulo n there are phi(n) different characters. [This is made explicit for example by the number of rows in arXiv:1008.2547.] The set of the fourth powers of the characters in all representations has some cardinality, which defines the sequence.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10000
R. J. Mathar, Size of the Set of Residues of Integer Powers of Fixed Exponent, (2017).
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010.
Wikipedia, Dirichlet character.
FORMULA
Multiplicative with a(2^e) = 1 for e<=3; a(2^e) = 2^(e-4) for e>=4; a(p^e) = p^(e-1)*(p-1)/4 for e>=1 and p == 1 (mod 4); a(p^e) = p^(e-1)*(p-1)/2 for e>=1 and p == 3 (mod 4). (Derived from A073103.) - R. J. Mathar, Oct 13 2017
EXAMPLE
For n <= 6, the set of all characters in all representations consists of a subset of +1, -1, +i or -i. Their fourth powers are all +1, a single value, so a(n)=1 then.
For n=7, the set of characters is 1, -1, +-1/2 +- sqrt(3)*i/2, so their fourth powers are 1 or -1/2 +- sqrt(3)*i/2, which are three different values, so a(7)=3.
For n=11, the fourth powers of the characters may be 1, exp(+-2*i*Pi/5) or exp(+-4*i*Pi/5), which are 5 different values.
MATHEMATICA
a[n_] := EulerPhi[n]/Count[Range[0, n-1]^4 - 1, k_ /; Divisible[k, n]];
Array[a, 80] (* Jean-François Alcover, Nov 20 2017 *)
f[p_, e_] := (p - 1)*p^(e - 1)/If[Mod[p, 4] == 1, 4, 2]; f[2, e_] := If[e <= 3, 1, 2^(e - 4)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 10 2023 *)
PROG
(PARI) a(n)=my(f=factor(n)); prod(i=1, #f~, if(f[i, 1]==2, 2^max(0, f[i, 2]-4), f[i, 1]^(f[i, 2]-1)*(f[i, 1]-1)/if(f[i, 1]%4==1, 4, 2))) \\ Charles R Greathouse IV, Mar 02 2015
CROSSREFS
KEYWORD
easy,nonn,mult
AUTHOR
R. J. Mathar, Mar 02 2015
STATUS
approved