A376427 The number of distinct values of x+y+z+w (mod n) when x*y*z*w = 1 (mod n).

1, 1, 3, 1, 5, 3, 7, 2, 5, 5, 11, 3, 13, 7, 15, 4, 17, 5, 19, 5, 21, 11, 23, 6, 25, 13, 15, 7, 29, 15, 31, 8, 33, 17, 35, 5, 37, 19, 39, 10, 41, 21, 43, 11, 25, 23, 47, 12, 49, 25, 51, 13, 53, 15, 55, 14, 57, 29, 59, 15, 61, 31, 35, 16, 65, 33, 67, 17, 69, 35, 71, 10, 73, 37, 75, 19, 77, 39, 79, 20, 45, 41, 83, 21, 85, 43, 87, 22, 89, 25

Offset: 1

Author

W. Edwin Clark, Sep 22 2024

Keywords

• nonn

Comments

The values of n for which a(n) = n seem to agree with A325128. But I have no proof.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..1688

Crossrefs

Cf. A376296, A376183, A180783.

Programs

Maple

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

Python

def A376427(n):
    s = set()
    for x in range(n):
        for y in range(x,n):
            xy, xyp = x*y%n, (x+y)%n
            for z in range(y,n):
                try:
                    s.add((xyp+z+pow(xy*z%n,-1,n))%n)
                except:
                    continue
    return len(s) # Chai Wah Wu, Sep 23 2024