A087412 a(n) is the number of solutions to x^3 + y^3 == 1 (mod n).

1, 2, 3, 4, 5, 6, 6, 8, 18, 10, 11, 12, 6, 12, 15, 16, 17, 36, 24, 20, 18, 22, 23, 24, 25, 12, 54, 24, 29, 30, 33, 32, 33, 34, 30, 72, 24, 48, 18, 40, 41, 36, 33, 44, 90, 46, 47, 48, 42, 50, 51, 24, 53, 108, 55, 48, 72, 58, 59, 60

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Formula

From Andrew Howroyd, Jul 17 2018: (Start)

a(p^e) = p^e for p prime and p mod 3 = 2.

Conjecture: a(3^e) = 2*3^e for e > 1.

a(p^e) = p^(e-1)*(p - 1 + Sum_{b=1..p-1} Legendre(12*b^(-1) - 3*b^2, p)) for p prime and p <> 3.

The final formula arises from factoring x^3 + y^3 as (x + y)*(x*2 - x*y + y^2), then substituting b = x + y and counting the solutions to the resulting quadratic equation 3*x^2 - 3*b*x + b^2 == b^(-1) (mod p) for each nonzero value of b. (End)

Mathematica

a[n_] := Module[{v = Table[0, {n}]}, For[i = 0, i <= n-1, i++, v[[Mod[i^3, n] + 1]]++]; Sum[v[[i+1]] v[[Mod[1-i, n] + 1]], {i, 0, n-1}]];
a /@ Range[1, 60] (* Jean-François Alcover, Sep 17 2019, after Andrew Howroyd *)

Prog

(PARI) a(n) = {nb = 0; for (x = 0, n-1, for (y = 0, n-1, if (Mod(x^3, n) + Mod(y^3, n) == Mod(1, n), nb++); ); ); nb; } \\ Michel Marcus, Aug 06 2013
(PARI) a(n)={my(v=vector(n)); for(i=0, n-1, v[i^3%n + 1]++); sum(i=0, n-1, v[i+1]*v[(1-i)%n + 1])} \\ Andrew Howroyd, Jul 17 2018

Crossrefs

Cf. A087786.

Sequence in context: A128559 A122168 A017856 * A024684 A265538 A037850

Adjacent sequences: A087409 A087410 A087411 * A087413 A087414 A087415

Keyword

mult,nonn

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 21 2003

Extensions

More terms from Michel Marcus, Aug 06 2013

Status

approved