A052274 -id:A052274 - cp's OEIS frontend

A228849 Number of distinct 12th powers modulo n.

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 6, 4, 2, 4, 4, 2, 5, 4, 4, 4, 4, 12, 12, 4, 6, 4, 4, 4, 8, 8, 6, 3, 12, 10, 4, 4, 4, 8, 4, 4, 11, 8, 8, 12, 4, 24, 24, 4, 8, 12, 10, 4, 14, 8, 12, 4, 8, 16, 30, 8, 6, 12, 4, 5, 4, 24, 12, 10, 24, 8, 36, 4, 7, 8, 12, 8, 12, 8, 14, 4, 10

Offset: 1

Arkadiusz Wesolowski, Sep 05 2013

R. J. Mathar, Table of n, a(n) for n = 1..10000

Cf. A000224 (squares), A046530 (cubic residues), A052273 (4th powers), A052274 (5th powers), A052275 (6th powers), A085310 (7th powers), A085311 (8th powers), A085312 (9th powers), A085313 (10th powers), A085314 (11th powers).

[#Set([k^12 mod n : k in [1..n]]) : n in [1..81]];

A228849 := proc(n)
    {seq(i^12 mod n, i=0..n-1)} ;
    nops(%) ;
end proc: # R. J. Mathar, Sep 21 2017

a[n_] := Table[PowerMod[i, 12, n], {i, 0, n - 1}] // Union // Length;
Array[a, 100] (* Jean-François Alcover, Mar 24 2020 *)

a(n)=my(f=factor(n)); prod(i=1, #f[, 1], my(k=f[i, 1]^f[i, 2]); #vecsort(vector(k, i, i^12%k), , 8)) \\ Charles R Greathouse IV, Sep 05 2013

A257303 Number of 5th power nonresidues modulo n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 3, 2, 0, 8, 3, 0, 0, 0, 7, 0, 4, 0, 5, 0, 16, 0, 9, 20, 0, 8, 7, 0, 0, 24, 15, 24, 0, 0, 15, 0, 0, 0, 15, 32, 0, 0, 35, 10, 0, 0, 21, 6, 40, 0, 13, 0, 16, 40, 21, 0, 0, 0, 15, 48, 48, 14, 30, 0, 48, 0, 17, 0, 0, 56, 37, 0, 0, 60, 19, 56, 0, 0, 35, 26, 64, 0, 21, 0, 0, 0, 73, 0

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), A257302 (p=4).

Table[Length[Complement[Range[n - 1], Union[Mod[Range[n]^5, 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,5)

a(n) = n-A052274(n).

Satisfies a(n) <= n-3 (residues 0, 1, and n-1 are always present).

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

A228849 Number of distinct 12th powers modulo n.

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A257303 Number of 5th power nonresidues modulo n.

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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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