A087688 a(n) = number of solutions to x^3 - x == 0 (mod n).

1, 2, 3, 3, 3, 6, 3, 5, 3, 6, 3, 9, 3, 6, 9, 5, 3, 6, 3, 9, 9, 6, 3, 15, 3, 6, 3, 9, 3, 18, 3, 5, 9, 6, 9, 9, 3, 6, 9, 15, 3, 18, 3, 9, 9, 6, 3, 15, 3, 6, 9, 9, 3, 6, 9, 15, 9, 6, 3, 27, 3, 6, 9, 5, 9, 18, 3, 9, 9, 18

Offset

1,2

Comments

Shadow transform of A007531. - Michel Marcus, Jun 06 2013

a(n) = 3 iff n belongs to (A061345 \ {1}) Union {4}. - Bernard Schott, Sep 16 2019

Links

Eric M. Schmidt, Table of n, a(n) for n = 1..10000

Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150; see Definition 7 for the shadow transform.

N. J. A. Sloane, Transforms.

Formula

Multiplicative with a(p^e) = 3 for p an odd prime, a(2^1) = 2, a(2^2) = 3, a(2^e) = 5 for e >= 3. - Eric M. Schmidt, Apr 08 2013

Maple

A087688 := proc(n) local a, x ; a := 0 ; for x from 0 to n-1 do if (x*(x^2-1)) mod n = 0 then a := a+1 ; end if; end do; a ; end proc:
seq(A087688(n), n=1..70) ; # R. J. Mathar, Jan 07 2011

Mathematica

nsols[n_]:=Length[Select[Range[0, n-1], Mod[#^3-#, n]==0&]]; nsols/@Range[80]  (* Harvey P. Dale, Mar 22 2011 *)
f[2, e_] := Which[e == 1, 2, e == 2, 3, e >= 3, 5]; f[p_, e_] := 3; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)

Prog

(PARI) a(n)=if(n%2, 3^omega(n), my(v=valuation(n, 2)); 3^omega(n>>v)*[2, 3, 5][min(3, v)]) \\ Charles R Greathouse IV, Mar 22 2011

Crossrefs

Cf. A034444, A224516, A060594.

Sequence in context: A134187 A078644 A133700 * A126854 A115206 A093653

Adjacent sequences: A087685 A087686 A087687 * A087689 A087690 A087691

Keyword

mult,nonn,easy

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 27 2003

Status

approved