A160499 -id:A160499 - cp's OEIS frontend

A160498 Number of cubic primitive Dirichlet characters modulo n.

1, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0

Offset: 1

Steven Finch, May 15 2009

Also called primitive Dirichlet characters of order 3.
Mobius transform of A060839.
C. David, J. Fearnley & H. Kisilevsky prove that Sum_{k=1..n} a(k) ~ C*n, with C = (11*sqrt(3)/(18*Pi)) * Product_{primes p == 1 (mod 3)} (1 - 2/(p*(p+1))) = 0.3170565167922841205670156...; they credit Cohen, F. Diaz y Diaz, & M. Olivier 2002 (see Proposition 5.2. and Corollary 5.3.). - Charles R Greathouse IV, Aug 26 2009 [corrected by Vaclav Kotesovec, Sep 16 2020]
a(n) is the number of primitive Dirichlet characters modulo n such that all entries are 0 or a cubic root of unity: 1, w = (-1 + sqrt(3)*i)/2 or w^2 = (-1 - sqrt(3)*i)/2. - Jianing Song, Feb 27 2019
Every term is 0 or a power of 2. - Jianing Song, Mar 02 2019
From Jianing Song, Apr 03 2021: (Start)
For n >= 2, a(n) is the number of cyclic cubic fields with discriminant n^2. See A343023 for detailed information.
The first occurrence of 2^t is 9*A121940(t-1) for t >= 2. (End)

			From _Jianing Song_, Mar 02 2019: (Start)
Let w = (-1 + sqrt(3)*i)/2 be one of the primitive 3rd root of unity.
For n = 7, the 2 cubic primitive Dirichlet characters modulo n are [0, 1, w, w^2, w^2, w, 1] and [0, 1, w^2, w, w, w^2, 1], so a(7) = 2.
For n = 9, the 2 cubic primitive Dirichlet characters modulo n are [0, 1, w, 0, w^2, w^2, 0, w, 1] and [0, 1, w^2, 0, w, w, 0, w^2, 1], so a(9) = 2. (End)

Jianing Song, Table of n, a(n) for n = 1..10000
C. David, J. Fearnley and H. Kisilevsky,On the vanishing of twisted L-functions of elliptic curves, Experim. Math. 13 (2004) 185-198.
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.
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n for n = 1..10000000

Cf. A114643 (number of quadratic primitive Dirichlet characters modulo n), A160499 (number of quartic primitive Dirichlet characters modulo n).
Cf. A060839 (number of solutions to x^3 == 1 (mod n)).
Cf. A121940, A343023.

A060839[n_] := Sum[If[Mod[k^3 - 1, n] == 0, 1, 0], {k, 1, n}]; a[n_] := Sum[ MoebiusMu[n/d]*A060839[d], {d, Divisors[n]}]; Table[a[n], {n, 2, 81}] (* Jean-François Alcover, Jun 19 2013 *)
f[3, 2] = 2; f[p_, e_] := If[Mod[p, 3] == 1 && e == 1, 2, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

a(n)=sum(d=1, n, if(n%d==0, moebius(n/d)*sum(i=1, d, if((i^3-1)%d, 0, 1)), 0)) \\ Steven Finch, Jun 09 2009

A005088(n)=my(f=factor(n)[,1]); sum(i=1,#f,f[i]%3==1)
A060839(n)=3^((n%9==0)+A005088(n))
a(n)=sumdiv(n,d,moebius(n/d)*A060839(d)) \\ Charles R Greathouse IV, Aug 26 2009

a(n) = my(L=factor(n), w=omega(n)); for(i=1, w, if(!((L[i, 1]%3==1 && L[i, 2]==1) || L[i, 1]^L[i, 2] == 9), return(0))); 2^w \\ Jianing Song, Apr 03 2021

Multiplicative with a(p^e) = 2 if p^e = 9 or p == 1 (mod 3) and e = 1, otherwise 0. - Jianing Song, Mar 02 2019

a(n) = 2*A343023(n) for n >= 2. - Jianing Song, Apr 03 2021

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

A114643 Number of real primitive Dirichlet characters modulo n.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 1, 2, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 2, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 2, 1

Offset: 1

Steven Finch, Feb 16 2006

a(n) = 1 if either n or -n is a fundamental discriminant (not both); a(n) = 2 if n and -n are fundamental discriminants; a(n) = 0 otherwise. Also, Sum_{k=1..n} a(k) is asymptotic to (6/Pi^2)*n.
From Jianing Song, Feb 27 2019: (Start)
If n is an odd squarefree number, then a(n) = 1, where the unique real primitive Dirichlet character modulo n is {Kronecker(n,k)} = {Jacobi(k,n)} if n == 1 (mod 4) and {Kronecker(-n,k)} = {Jacobi(k,n)} if n == 3 (mod 4).
If n = 4*m, m is an odd squarefree number, then a(n) is also 1, where the unique real primitive Dirichlet character modulo n is {Kronecker(-n,k)} if m == 1 (mod 4) and {Kronecker(n,k)} if m == 3 (mod 4).
If n is 8 times an odd squarefree number, then a(n) = 2, where the two real primitive Dirichlet characters modulo n are {Kronecker(n,k)} and {Kronecker(-n,k)}.
a(n) = 0 if n == 2 (mod 4), n is divisible by 16 or the square of an odd prime. (End)
Mobius transform of A060594. - Jianing Song, Mar 02 2019

			From _Jianing Song_, Feb 27 2019: (Start)
For n = 5, the only real primitive Dirichlet characters modulo n is {Kronecker(5,k)} = [0, 1, -1, -1, 1] = A080891, so a(5) = 1.
For n = 8, the real primitive Dirichlet characters modulo n are {Kronecker(8,k)} = [0, 1, 0, -1, 0, -1, 0, 1] = A091337 and [0, 1, 0, 1, 0, -1, 0, -1] = A188510, so a(8) = 2.
For n = 20, the only real primitive Dirichlet characters modulo n is {Kronecker(-20,k)} = [0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1] = A289741, so a(20) = 1. (End)

W. Ellison and F. Ellison, Prime Numbers, Wiley, 1985, pp. 224-226.

Robert Israel, 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.
Vaclav Kotesovec, Graph - the asymptotic ratio
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
I. J. Zucker and M. M. Robertson, Some properties of Dirichlet L-series, J. Phys. A 9 (1976) 1207-1214.

Cf. A003657, A003658.
Cf. A160498 (number of cubic primitive Dirichlet characters modulo n), A160499 (number of quartic primitive Dirichlet characters modulo n).
Cf. A060594 (number of solutions to x^2 == 1 (mod n)).

A114643 := proc(n)
    local a,pf,p,r;
    a := 1 ;
    for pf in ifactors(n)[2] do
        p := op(1,pf);
        r := op(2,pf);
        if p = 2 then
            if r =  1 then
                a := 0 ;
            elif r =  2 then
                ;
            elif r =  3 then
                a := a*2 ;
            elif r >=  4 then
                a := 0 ;
            end if;
        else
            if r =1 then
                ;
            else
                a := 0 ;
            end if;
        end if;
    end do:
    a ;
end proc:
seq(A114643(n),n=1..40) ; # R. J. Mathar, Mar 02 2015
# Alternative:
f:= proc(n) local r,v,F;
  v:= padic:-ordp(n,2);
  if v = 1 or v >= 4 then return 0
  elif v = 3 then r:= 2
  else r:= 1
  fi;
  if numtheory:-issqrfree(n/2^v) then r else 0 fi
end proc:
map(f, [$1..100]); # Robert Israel, Oct 08 2017

a[n_] := Sum[ MoebiusMu[n/d] * Sum[ If[ Mod[i^2 - 1, d] == 0, 1, 0], {i, 2, d}], {d, Divisors[n]}]; a[1] = 1; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Jun 20 2013, after Steven Finch *)
f[2, e_] := Which[e == 1, 0, e == 2, 1, e == 3, 2, e >= 4, 0]; f[p_, e_] := If[e == 1, 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

a(n)=sum(d=1, n, if(n%d==0, moebius(n/d)*sum(i=1, d, if((i^2-1)%d, 0, 1)), 0)) \\ Steven Finch, Jun 09 2009

This sequence is multiplicative with a(2) = 0, a(4) = 1, a(8) = 2, a(2^r) = 0 for r > 3, a(p) = 1 for prime p > 2 and a(p^r) = 0 for r > 1. - Steven Finch, Mar 08 2006 (With correction by Jianing Song, Jun 28 2018)

Dirichlet g.f.: zeta(s)*(1 + 2^(-2s) + 2^(1-3s))/(zeta(2s)*(1 + 2^(-s))). - R. J. Mathar, Jul 03 2011

A307380 Number of quintic primitive Dirichlet characters modulo n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 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, 4, 0, 0, 0, 0

Offset: 1

Jianing Song, Apr 06 2019

a(n) is the number of primitive Dirichlet characters modulo n such that all entries are 0 or a fifth-power root of unity.

Mobius transform of A319099. Every term is 0 or a power of 4.

			Let w = exp(2*Pi/5). For n = 11, the 4 quintic primitive Dirichlet characters modulo n are:
  Chi_1 = [0, 1, w, w^3, w^2, w^4, w^4, w^2, w^3, w, 1];
  Chi_2 = [0, 1, w^2, w, w^4, w^3, w^3, w^4, w, w^2, 1];
  Chi_3 = [0, 1, w^3, w^4, w, w^2, w^2, w, w^4, w^3, 1];
  Chi_4 = [0, 1, w^4, w^2, w^3, w, w, w^3, w^2, w^4, 1],
so a(11) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65539

Number of k-th power primitive Dirichlet characters modulo n: A114643 (k=2), A160498 (k=3), A160499 (k=4), this sequence (k=5), A307381 (k=6), A307382 (k=7), A329272 (k=8).

Cf. A319099 (number of solutions to x^5 == 1 (mod n)).

f[5, 2] = 4; f[p_, e_] := If[Mod[p, 5] == 1 && e == 1, 4, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

a(n)=sum(d=1, n, if(n%d==0, moebius(n/d)*sum(i=1, d, if((i^5-1)%d, 0, 1)), 0))

A307380(n) = sumdiv(n, d, moebius(n/d)*sum(i=1, d, if((i^5-1)%d, 0, 1))); \\ (Slightly speeding the program above) - Antti Karttunen, Aug 22 2019

A307380(n) = { my(f=factor(n)); prod(i=1, #f~, if(((5==f[i,1])&&(2==f[i,2]))||((1==(f[i,1]%5))&&(1==f[i,2])),4,0)); }; \\ (After the multiplicative formula, much faster) - Antti Karttunen, Aug 22 2019

Multiplicative with a(p^e) = 4 if p^e = 25 or p == 1 (mod 5) and e = 1, otherwise 0.

More terms from Antti Karttunen, Aug 22 2019

A307381 Number of sextic primitive Dirichlet characters modulo n.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 5, 2, 4, 0, 1, 1, 5, 0, 1, 0, 1, 0, 5, 1, 5, 0, 1, 2, 0, 0, 0, 5, 1, 0, 5, 0, 1, 0, 5, 4, 5, 0, 5, 2, 1, 0, 5, 1, 4, 0, 1, 0, 0, 0, 1, 5, 1, 0, 1, 10, 5, 0, 1, 1, 5, 0, 20, 0, 5, 0, 5, 1, 1, 0, 1, 8, 5, 0, 0, 5, 5, 0, 5, 0, 0, 0, 1, 5, 1, 0, 1

Offset: 1

Jianing Song, Apr 06 2019

a(n) is the number of primitive Dirichlet characters modulo n such that all entries are 0 or a six-power root of unity (1, (1 + sqrt(3)*i)/2, (-1 + sqrt(3)*i)/2, -1, (-1 - sqrt(3)*i)/2, (1 - sqrt(3)*i)/2).

Mobius transform of A319100.

			Let w = exp(2*Pi/6) = (1 + sqrt(3)*i)/2. For n = 19, the 5 sextic primitive Dirichlet characters modulo n are:
  Chi_1 = [0, 1, w, w, w - 1, -w, w - 1, 1, -1, w - 1, -w + 1, 1, -1, -w + 1, w, -w + 1, -w, -w, -1];
  Chi_2 = [0, 1, w - 1, w - 1, -w, w - 1, -w, 1, 1, -w, -w, 1, 1, -w, w - 1, -w, w - 1, w - 1, 1];
  Chi_3 = [0, 1, -1, -1, 1, 1, 1, 1, -1, 1, -1, 1, -1, -1, -1, -1, 1, 1, -1];
  Chi_4 = [0, 1, -w, -w, w - 1, -w, w - 1, 1, 1, w - 1, w - 1, 1, 1, w - 1, -w, w - 1, -w, -w, 1];
  Chi_5 = [0, 1, -w + 1, -w + 1, -w, w - 1, -w, 1, -1, -w, w, 1, -1, w, -w + 1, w, w - 1, w - 1, -1],
so a(19) = 5.

Jianing Song, Table of n, a(n) for n = 1..65539

Number of k-th power primitive Dirichlet characters modulo n: A114643 (k=2), A160498 (k=3), A160499 (k=4), A307380 (k=5), this sequence (k=6), A307382 (k=7), A329272 (k=8).

Cf. A319100 (number of solutions to x^6 == 1 (mod n)).

f[2, e_] := Which[e == 1, 0, e == 2, 1, e == 3, 2, e >= 4, 0]; f[3, e_] := Which[e == 1, 1, e == 2, 4, e >= 3, 0]; f[p_, 1] := If[Mod[p, 6] == 1, 5, 1]; f[p_, e_] := 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

a(n)={
    my(r=1, f=factor(n));
    for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);
        if(p==2, if(e==3, r*=2, if(e!=2, r=0; return(r))));
        if(p==3, if(e==2, r*=4, if(e!=1, r=0; return(r))));
        if(p>3, if(p%6==1&&e==1, r*=5, if(e!=1, r=0; return(r))));
    );
    return(r);
} \\ Jianing Song, Nov 10 2019

Multiplicative with a(4) = 1, a(8) = 2, a(2^e) = 0 for e = 1 or e >= 4; a(3) = 1, a(9) = 4, a(3^e) = 0 for e >= 3; a(p) = 5 if p == 1 (mod 6) and 1 if p == 5 (mod 6), a(p^e) = 0 if p > 3 and e >= 2.

A307382 Number of septic primitive Dirichlet characters modulo n.

Original entry on oeis.org

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

Jianing Song, Apr 06 2019

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.

			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.

Antti Karttunen, Table of n, a(n) for n = 1..65537

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)).

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 *)

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))

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

Multiplicative with a(p^e) = 6 if p^e = 49 or p == 1 (mod 7) and e = 1, otherwise 0.

More terms from Antti Karttunen, Aug 22 2019

A329272 Number of octic primitive Dirichlet characters modulo n.

Original entry on oeis.org

1, 0, 1, 1, 3, 0, 1, 2, 0, 0, 1, 1, 3, 0, 3, 4, 7, 0, 1, 3, 1, 0, 1, 2, 0, 0, 0, 1, 3, 0, 1, 8, 1, 0, 3, 0, 3, 0, 3, 6, 7, 0, 1, 1, 0, 0, 1, 4, 0, 0, 7, 3, 3, 0, 3, 2, 1, 0, 1, 3, 3, 0, 0, 0, 9, 0, 1, 7, 1, 0, 1, 0, 7, 0, 0, 1, 1, 0, 1, 12, 0, 0, 1, 1, 21, 0, 3, 2, 7, 0, 3

Offset: 1

Jianing Song, Nov 10 2019

a(n) is the number of primitive Dirichlet characters modulo n such that all entries are 0 or a eighth-power root of unity (+-1, +-i, +-sqrt(2)/2 +- sqrt(2)/2*i, i = sqrt(-1)).

Mobius transform of A247257.

			Let w = exp(2*Pi*i/8) = sqrt(2)/2 + i*sqrt(2)/2. For n = 17, the 7 octic primitive Dirichlet characters modulo n are:
  Chi_1 = [0, 1, -i, w, -1, -w, -w^3, w^3, i, i, w^3, -w^3, -w, -1, w, -i, 1];
  Chi_2 = [0, 1, -1, i, 1, i, -i, -i, -1, -1, -i, -i, i, 1, i, -1, 1];
  Chi_3 = [0, 1, i, w^3, -1, -w^3, -w, w, -i, -i, w, -w, -w^3, -1, w^3, i, 1];
  Chi_4 = [0, 1, 1, -1, 1, -1, -1, -1, 1, 1, -1, -1, -1, 1, -1, 1, 1];
  Chi_5 = [0, 1, -i, -w, -1, w, w^3, -w^3, i, i, -w^3, w^3, w, -1, -w, -i, 1];
  Chi_6 = [0, 1, -1, -i, 1, -i, i, i, -1, -1, i, i, -i, 1, -i, -1, 1];
  Chi_7 = [0, 1, i, -w^3, -1, w^3, w, -w, -i, -i, -w, w, w^3, -1, -w^3, i, 1],
so a(17) = 7.

Jianing Song, Table of n, a(n) for n = 1..65539

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), A307382 (k=7), this sequence (k=8).

Cf. A247257 (number of solutions to x^8 == 1 (mod n)).

f[2, e_] := If[2 <= e <= 5, 2^(e-2), 0]; f[p_, e_] := If[e == 1, GCD[p-1, 8] - 1, 0];  a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

a(n)={
    my(r=1, f=factor(n));
    for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);
        if(p==2, if(e>=2&&e<=5, r*=2^(e-2), r=0; return(r)));
        if(p>2, if(e==1, r*=gcd(p-1,8)-1, r=0; return(r)));
    );
    return(r);
}

Multiplicative with a(2^e) = 2^(e-2) for 2 <= e <= 5, a(2^e) = 0 for e = 1 or e >= 6; a(p^e) = gcd(p-1, 8)-1 if p > 2 and e = 1, a(p^e) = 0 if p > 2 and e >= 2.

A354058 Square array read by ascending antidiagonals: T(n,k) is the number of degree-k primitive Dirichlet characters modulo n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 2, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 2, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 5, 0, 3, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 1

Offset: 1

Jianing Song, May 16 2022

Given n, T(n,k) only depends on gcd(k,psi(n)). For the truncated version see A354061.
Each column is multiplicative.
The n-th rows contains entirely 0's if and only if n == 2 (mod 4).
For n !== 2 (mod 4), T(n,psi(n)) > T(n,k) if k is not divisible by psi(n).
Proof: this is true if n is a prime power (see the formula below). Now suppose that n = Product_{i=1..r} (p_i)^(e_i). Since n !== 2 (mod 4), (p_i)^(e_i) != 2, so T((p_i)^(e_i),psi((p_i)^(e_i))) > 0 for each i. If k is not divisible by psi(n), then it is not divisible by some psi((p_{i_0})^(e_{i_0})), so T(n,psi(n)) = Product_{i=1..r} T((p_i)^(e_i),psi(n)) = Product_{i=1..r} T((p_i)^(e_i),psi((p_i)^(e_i))) > T((p_{i_0})^(e_{i_0}),k) * Product_{i!=i_0} T((p_i)^(e_i),psi((p_i)^(e_i))) >= Product_{i=1..r} T((p_i)^(e_i),k) = T(n,k).

			  n/k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20
   1   1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
   2   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   3   0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1
   4   0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1
   5   0  1  0  3  0  1  0  3  0  1  0  3  0  1  0  3  0  1  0  3
   6   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   7   0  1  2  1  0  5  0  1  2  1  0  5  0  1  2  1  0  5  0  1
   8   0  2  0  2  0  2  0  2  0  2  0  2  0  2  0  2  0  2  0  2
   9   0  0  2  0  0  4  0  0  2  0  0  4  0  0  2  0  0  4  0  0
  10   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
  11   0  1  0  1  4  1  0  1  0  9  0  1  0  1  4  1  0  1  0  9
  12   0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1
  13   0  1  2  3  0  5  0  3  2  1  0 11  0  1  2  3  0  5  0  3
  14   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
  15   0  1  0  3  0  1  0  3  0  1  0  3  0  1  0  3  0  1  0  3
  16   0  0  0  4  0  0  0  4  0  0  0  4  0  0  0  4  0  0  0  4
  17   0  1  0  3  0  1  0  7  0  1  0  3  0  1  0 15  0  1  0  3
  18   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
  19   0  1  2  1  0  5  0  1  8  1  0  5  0  1  2  1  0 17  0  1
  20   0  1  0  3  0  1  0  3  0  1  0  3  0  1  0  3  0  1  0  3

Jianing Song, Table of n, a(n) for n = 1..5050 (the first 100 ascending diagonals)

k-th column: A114643 (k=2), A160498 (k=3), A160499 (k=4), A307380 (k=5), A307381 (k=6), A307382 (k=7), A329272 (k=8).
Moebius transform of A354057 applied to each column.
A354257 gives the smallest index for the nonzero terms in each row.
Cf. A007431.

b(n,k)=my(Z=znstar(n)[2]); prod(i=1, #Z, gcd(k, Z[i]));
T(n,k) = sumdiv(n, d, moebius(n/d)*b(d,k))

For odd primes p: T(p,k) = gcd(p-1,k)-1, T(p^e,k*p^(e-1)) = p^(e-2)*(p-1)*gcd(k,p-1), T(p^e,k) = 0 if k is not divisible by p^(e-1). T(2,k) = 0, T(4,k) = 1 for even k and 0 for odd k, T(2^e,k) = 2^(e-2) if k is divisible by 2^(e-2) and 0 otherwise.

T(n,psi(n)) = A007431(n). - Jianing Song, May 24 2022

cp's OEIS Frontend

A160498 Number of cubic primitive Dirichlet characters modulo n.

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A307380 Number of quintic primitive Dirichlet characters modulo n.

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A307381 Number of sextic primitive Dirichlet characters modulo n.

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A307382 Number of septic primitive Dirichlet characters modulo n.

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A329272 Number of octic primitive Dirichlet characters modulo n.

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