A052273 -id:A052273 - cp's OEIS frontend

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

R. J. Mathar, Mar 02 2015

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.

			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.

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.

Cf. A046073, A087692, A052273, A293482 - A293485.

A250207 := proc(n)
    numtheory[phi](n)/A073103(n) ;
end proc:

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 *)

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

a(n) = A000010(n)/A073103(n).

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

A257302 Number of 4th power nonresidues modulo n.

Original entry on oeis.org

0, 0, 1, 2, 3, 2, 3, 6, 5, 6, 5, 8, 9, 6, 11, 14, 12, 10, 9, 16, 13, 10, 11, 20, 19, 18, 17, 20, 21, 22, 15, 28, 21, 24, 27, 28, 27, 18, 31, 36, 30, 26, 21, 32, 37, 22, 23, 44, 27, 38, 41, 44, 39, 34, 43, 48, 37, 42, 29, 52, 45, 30, 47, 58, 57, 42, 33, 58

Offset: 1

Stanislav Sykora, Apr 19 2015

nonn

a(n) is the number of values r, 0<=r=0, (m^p)%n != r.

Stanislav Sykora, Table of n, a(n) for n = 1..10000

Cf. A095972 (p=2), A257301 (p=3), A257303 (p=5).

Table[Length[Complement[Range[n - 1], Union[Mod[Range[n]^4, n]]]], {n, 100}] (* Vincenzo Librandi, Apr 20 2015 *)

nrespowp(n,p) = {my(v=vector(n),d=0);
  for(r=0,n-1,v[1+(r^p)%n]+=1);
  for(k=1,n,if(v[k]==0,d++));
  return(d);}
a(n) = nrespowp(n,4)

a(n) = n - A052273(n).

Satisfies a(n) <= n-2 (residues 0 and 1 are always present).

A096094 Analog of A094091 for S=3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1

Offset: 1

Joshua Zucker, Jul 23 2006

A finite sequence of length 23.

The old entry with this A-number was a duplicate of A052273.

Cf. A094091, A106197.

A337868 Number of distinct residues of x^r (mod n), x=0..n-1, r=2, ..., n.

Original entry on oeis.org

0, 2, 3, 3, 5, 6, 7, 6, 7, 10, 11, 9, 13, 14, 15, 11, 17, 14, 19, 15, 21, 22, 23, 17, 21, 26, 20, 21, 29, 30, 31, 21, 33, 34, 35, 21, 37, 38, 39, 28, 41, 42, 43, 33, 35, 46, 47, 32, 43, 42, 51, 39, 53, 40, 55, 39, 57, 58, 59, 45, 61, 62, 49, 41, 65, 66, 67, 51, 69, 70, 71

Offset: 1

José María Grau Ribas, Sep 27 2020

nonn

Sequence is submultiplicative: a(m*n) <= a(m) * a(n) for m,n coprime. - Charles R Greathouse IV, Dec 19 2022

For n > 1, this is the number of distinct residues of x^r (mod n) with r > 1, that is, the restriction r <= n is not needed. - Charles R Greathouse IV, Dec 22 2022

For number of k-th power residues mod n, see A000224 (k=2), A052273 (k=4), A052274 (k=5), A052275 (k=6), A085310 (k=7), A085311 (k=8), A085312 (k=9), A085313 (k=10), A085314 (k=12), A228849 (k=13).

T[n_] := Union@Mod[Flatten@Table[Range[n]^i, {i, 2, n}], n];
Table[Length[T@n], {n, 1, 144}]

a(n)=if(n==1, return(0)); my(s); for(k=0,n-1, my(x=Mod(k,n)); forprime(p=2,n, if(ispower(x,p), s++; break))); s\\ Charles R Greathouse IV, Dec 22 2022

For n > 1, a(n) >= A000010(n) + 1 as all invertible elements of Z/nZ are powers, as is 0. (Conjecture: equality holds exactly for A000430, the primes and squares of primes.) - Charles R Greathouse IV, Dec 23 2022

cp's OEIS Frontend

A250207 The number of quartic terms in the multiplicative group modulo n.

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A257302 Number of 4th power nonresidues modulo n.

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A096094 Analog of A094091 for S=3.

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A337868 Number of distinct residues of x^r (mod n), x=0..n-1, r=2, ..., n.

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