A376183 The number of solutions x<=y<=z in Z/(n) of the equation x+y+z = x*y*z.

1, 3, 2, 4, 7, 8, 10, 13, 13, 31, 24, 20, 37, 44, 38, 47, 59, 59, 66, 86, 53, 108, 96, 77, 137, 171, 100, 120, 159, 186, 170, 179, 135, 279, 230, 172, 253, 312, 220, 337, 307, 259, 322, 306, 331, 456, 384, 303, 369, 669, 366, 500, 503, 488, 588, 469, 409, 767, 600

Offset: 1

Author

W. Edwin Clark, Sep 14 2024

Keywords

• nonn

Comments

Suggested by a discussion initiated by Keith F. Lynch on the MathFun mailing list Sept 8, 2024 about when sums and products of real numbers x,y,z are integers and later raising other similar questions.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..8450 (terms 1..2000 from Hugo Pfoertner)

Programs

Maple

a:=proc(n)
local x,y,z,w,N;
N:=0:
for x from 0 to n-1 do
 for y from x to n-1 do
  for z from y to n-1 do
   if (x+y+z-x*y*z) mod n = 0 then N:=N + 1; fi;
  od:
 od:
od:
N;
end:

PARI

a(n) = sum(x=0, n-1, sum(y=x, n-1, sum(z=y, n-1, Mod(x+y+z-x*y*z, n)==0))); \\ Michel Marcus, Sep 15 2024

Python

def A376183(n):
    c = 0
    for x in range(n):
        for y in range(x,n):
            xy,xyp = x*y%n-1,(x+y)%n
            c += sum(not (xy*z-xyp)%n for z in range(y,n))
    return c # Chai Wah Wu, Sep 19 2024

Extensions

More terms from Hugo Pfoertner, Sep 15 2024