A087688 -id:A087688 - cp's OEIS frontend

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

Eric M. Schmidt, Apr 09 2013

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

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

Cf. A034444, A087688, A060839, A000086.

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 *)

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

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.

A087782 a(n) = number of solutions to x^3 + x == 0 (mod n).

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 1, 1, 6, 1, 1, 3, 2, 3, 1, 3, 2, 1, 3, 1, 2, 1, 1, 3, 6, 1, 1, 3, 6, 1, 1, 1, 6, 3, 1, 3, 2, 3, 3, 3, 2, 1, 1, 3, 2, 1, 1, 1, 6, 3, 3, 3, 2, 3, 1, 1, 6, 1, 3, 3, 2, 1, 1, 9, 2, 1, 3, 1, 6, 1, 1, 3, 6, 3, 1, 1, 6, 1, 3, 1, 6, 1, 1, 9, 2, 3, 1, 3, 6, 3, 1, 1, 2, 3, 1, 3, 2, 1, 3, 3, 6, 1, 3, 3

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 06 2003

Shadow transform of A034262. - Michel Marcus, Jun 06 2013

Andrew Howroyd, 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.

Cf. A087688, A000089, A034262.

a[n_] := If[n == 1, 1, Product[{p, e} = pe; If[p == 2, If[e == 1, 2, 1], If[Mod[p, 4] == 1, 3, 1]], {pe, FactorInteger[n]}]];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 20 2019 *)

a(n)={my(v=vector(n)); sum(i=0, n-1, lift(Mod(i,n)^3 + i) == 0)} \\ Andrew Howroyd, Jul 15 2018

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); if(p==2, if(e==1, 2, 1), if(p%4==1, 3, 1)))} \\ Andrew Howroyd, Jul 15 2018

Multiplicative with a(2^1) = 2, a(2^e) = 1 for e > 1, a(p^e) = 3 for p mod 4 == 1, a(p^e) = 1 for p mod 4 == 3. - Andrew Howroyd, Jul 15 2018

More terms from David Wasserman, Jun 17 2005

cp's OEIS Frontend

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

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