A132213 -id:A132213 - cp's OEIS frontend

A202034 Number of distinct prime residues of k^n (mod n), k=0..n-1.

0, 0, 1, 0, 2, 1, 3, 0, 0, 1, 4, 0, 5, 3, 6, 0, 6, 0, 7, 1, 2, 3, 8, 0, 1, 4, 0, 0, 9, 1, 10, 0, 11, 4, 11, 0, 11, 6, 3, 0, 12, 1, 13, 2, 3, 7, 14, 0, 2, 0, 15, 2, 15, 0, 3, 0, 5, 6, 16, 0, 17, 8, 0, 0, 18, 3, 18, 2, 19, 2, 19, 0, 20, 10, 2, 4, 21, 1, 21, 0, 0

Offset: 1

Michel Lagneau, Dec 09 2011

nonn

If n is a prime number, a(n) = A000720(n) - 1 because the number of distinct residues of k^n (mod n) = n.

			a(7) = 3  because  k^7 == 0, 1, 2, 3, 4, 5, 6 (mod 7) including 3 prime residues  2, 3, 5.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000720, A132213, A195637.

for n from 1 to 100 do: W:={}:for k from 0 to n-1 do:z:= irem(k^n,n): if type(z,prime)=true then W:=W union {z}:else fi:od: x:=nops(W): printf(`%d, `,x): od:

Table[Length[Select[Union[Table[Mod[k^n, n], {k, 0, n - 1}]], PrimeQ]], {n, 81}] (* Alonso del Arte, Dec 10 2011 *)
Count[Union[#],?PrimeQ]&/@Table[PowerMod[k,n,n],{n,100},{k,0,n-1}] (* _Harvey P. Dale, Sep 24 2022 *)

A132385 Number of distinct primes among the cubes mod n.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Nov 07 2007

This is to cubes A000578 as A132213 is to squares A000290.
It seems that the size of a(n) as compared to its surrounding elements is dependent on whether or not n is in A088232. If n is in A088232 the sequence assumes "big" values, otherwise the values will be "small". - Stefan Steinerberger, Nov 24 2007
If n is in A088232, a(n) = A000720(n-1) - A056170(n). - Robert Israel, Jun 28 2018

			a(10) = 4 because the cubes mod 10 repeat 0, 1, 8, 7, 4, 5, 6, 3, 2, 9, 0, 1, 8, 7, 4, 5, ... of which the 4 distinct primes are {2, 3, 5, 7}.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A000578, A000720, A056170, A132213.

f:= proc(n)
  if numtheory:-phi(n) mod 3 = 0 then nops(select(isprime, {seq(i^3 mod n, i=0..n-1)}))
  else numtheory:-pi(n-1) - nops(select(t -> t[2]>1, ifactors(n)[2]))
  fi
end proc:
map(f, [$1..100]); # Robert Israel, Jun 28 2018

Table[Length[Select[Union[Table[Mod[i^3, n], {i, 0, n}], Table[Mod[i^3, n], {i, 0, n}]], PrimeQ[ # ] &]], {n, 1, 70}] (* Stefan Steinerberger, Nov 12 2007 *)

a(n) = Card{p = k^3 mod n, for primes p and for all integers k}.

More terms from Stefan Steinerberger, Nov 12 2007

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010

cp's OEIS Frontend

A202034 Number of distinct prime residues of k^n (mod n), k=0..n-1.

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