A132385 Number of distinct primes among the cubes mod n.

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

1,5

Comments

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

Links

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

Formula

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

Example

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}.

Maple

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

Mathematica

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 *)

Crossrefs

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

Sequence in context: A339674 A241070 A128621 * A358840 A191716 A089235

Adjacent sequences: A132382 A132383 A132384 * A132386 A132387 A132388

Keyword

easy,nonn,look

Author

Jonathan Vos Post, Nov 07 2007

Extensions

More terms from Stefan Steinerberger, Nov 12 2007

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

Status

approved