A250207 The number of quartic terms in the multiplicative group modulo n.
1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 5, 1, 3, 3, 1, 1, 4, 3, 9, 1, 3, 5, 11, 1, 5, 3, 9, 3, 7, 1, 15, 2, 5, 4, 3, 3, 9, 9, 3, 1, 10, 3, 21, 5, 3, 11, 23, 1, 21, 5, 4, 3, 13, 9, 5, 3, 9, 7, 29, 1, 15, 15, 9, 4, 3, 5, 33, 4, 11, 3, 35, 3, 18, 9, 5, 9, 15, 3, 39, 1
Offset: 1
Examples
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.
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.
Programs
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Maple
A250207 := proc(n) numtheory[phi](n)/A073103(n) ; end proc:
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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 *)
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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
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
Comments