A224516 Number of solutions to x^4 - x == 0 (mod n).

1, 2, 2, 2, 2, 4, 4, 2, 4, 4, 2, 4, 4, 8, 4, 2, 2, 8, 4, 4, 8, 4, 2, 4, 2, 8, 4, 8, 2, 8, 4, 2, 4, 4, 8, 8, 4, 8, 8, 4, 2, 16, 4, 4, 8, 4, 2, 4, 4, 4, 4, 8, 2, 8, 4, 8, 8, 4, 2, 8, 4, 8, 16, 2, 8, 8, 4, 4, 4, 16, 2, 8, 4, 8, 4, 8, 8, 16, 4, 4, 4, 4, 2, 16, 4

Offset: 1

Author

Eric M. Schmidt, Apr 09 2013

Keywords

• nonn
• mult
• easy

Examples

			The solutions for n = 7 are 0, 1, 2, and 4.
		

Links

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

Crossrefs

Cf. A034444, A087688, A060839, A000086.

Programs

Mathematica

f[3, e_] := If[e == 1, 2, 4]; f[p_, e_] := If[Mod[p, 3] == 2, 2, 4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)

Sage

def A224516(n) :
    res = 1
    for p, m in factor(n) :
        if (p % 3 == 2) or (p == 3 and m == 1) : res *= 2
        else : res *= 4
    return res

Formula

Multiplicative with a(p^e) = 4 for p == 1 (mod 3); a(p^e) = 2 for p == 2 (mod 3); a(3^1) = 2; a(3^e) = 4 for e > 1.