A376183 -id:A376183 - cp's OEIS frontend

A376296 The number of solutions x<=y<=z<=w in Z/(n) of the equation x+y+z+w = xyz*w.

1, 2, 6, 7, 14, 18, 27, 34, 51, 59, 91, 96, 134, 136, 208, 203, 285, 261, 385, 373, 493, 487, 650, 616, 818, 750, 949, 947, 1240, 1146, 1517, 1397, 1766, 1662, 2089, 1824, 2443, 2309, 2723, 2638, 3311, 2977, 3801, 3482, 4024, 3962, 4900, 4382, 5525, 5023, 6078

Offset: 1

W. Edwin Clark, Sep 19 2024

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..1351

Cf. A376183, A180783.

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

def A376296(n):
    c = 0
    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):
                xyz, xyzp = xy*z%n-1,(xyp+z)%n
                c += sum(not (xyz*w-xyzp)%n for w in range(z,n))
    return c # Chai Wah Wu, Sep 19 2024

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

Original entry on oeis.org

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

W. Edwin Clark, Sep 22 2024

nonn

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

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

Cf. A376296, A376183, A180783.

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:

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

A376240 a(n) = number of {x,y,z} in Z/nZ such that x+y+z = xyz is nonzero.

Original entry on oeis.org

0, 1, 0, 1, 4, 2, 6, 6, 6, 21, 18, 7, 30, 30, 22, 28, 50, 37, 56, 63, 30, 86, 84, 42, 120, 145, 72, 87, 144, 124, 154, 140, 98, 245, 188, 123, 234, 274, 176, 274, 286, 169, 300, 253, 270, 410, 360, 196, 336, 607, 308, 437, 476, 388, 520, 378, 344, 709, 570, 429, 630, 758, 372, 600, 882, 517, 736, 751, 524, 968, 828, 546, 900, 1153, 810, 839, 720, 912, 1026, 1140, 738, 1413, 1134, 585, 1502, 1482, 930

Offset: 1

M. F. Hasler, Sep 24 2024

nonn

Similar to A376183, but the sum equal to the product is restricted to be nonzero, as to exclude many rather trivial solutions.

Counting sets {x, y, z} means that the order of the three (not necessarily distinct) values is irrelevant. Equivalently, one could use representatives with, e.g., x <= y <= z.

			For n=1, we can't have x+y+z = x*y*z nonzero because zero is the only element, so there are a(1) = 0 solutions.
For n=2, x+y+z = x*y*z != 0 implies x = y = z = 1 (in Z/2Z), so there is only one unique solution, and a(2) = 1.
For n=3, x+y+z = x*y*z != 0 is impossible: if x = y = z, the sum is zero (in Z/3Z), and 1 + 2 + z = 1*2*z <=> z = 0, so there is no solution, a(3) = 0.
For n=4, since 2*2 = 0 in Z/4Z, at most one among {x, y, z}, say z, can equal 2. In this case, x = -y = +-1, gives a solution {x, y, z} = {1, 2, 3}. One then checks that x = +-y = z = +-1 can't yield a solution, so a(4) = 1.
For n=5, we see that (x, y) = (2, 3) gives z = z, so we have a solution for any nonzero z, and exhaustive verification shows that these are no other solutions: a(5) = 4.

Chai Wah Wu, Table of n, a(n) for n = 1..7542

Cf. A376183 (same without restriction to nonzero sum/product).

apply( {A376240(n)=sum(x=1,n-1,sum(y=1,x,sum(z=1,y, (x+y+z-x*y*z)%n==0 && x*y*z%n)))}, [1..99])

def A376240(n):
    c = 0
    for x in range(1,n):
        for y in range(x,n):
            xy,xyp = x*y%n-1,n-(x+y)%n
            c += sum(not (z==xyp or (xy*z+xyp)%n) for z in range(y,n))
    return c # Chai Wah Wu, Sep 30 2024

A376318 The number of distinct values of x+y+z (mod n) when xyz = 1 (mod n).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 7, 2, 4, 4, 11, 2, 13, 7, 8, 3, 17, 4, 19, 4, 14, 11, 23, 4, 20, 13, 11, 7, 29, 8, 31, 6, 22, 17, 28, 4, 37, 19, 26, 8, 41, 14, 43, 11, 16, 23, 47, 6, 49, 20, 34, 13, 53, 11, 44, 14, 38, 29, 59, 8, 61, 31, 28, 11, 52, 22, 67, 17, 46, 28, 71, 8, 73, 37, 40, 19, 77, 26

Offset: 1

W. Edwin Clark, Sep 22 2024

nonn

The values of n for which a(n) = n seem to be A007775, but I have no proof of this.

Charles R Greathouse IV, Table of n, a(n) for n = 1..30000
Charles R Greathouse IV, Program for computing terms (C with PARI)

Cf. A007775, A376296, A376183, A180783, A376427.

a:=proc(n)
local x,y,z,N;
N:=NULL;
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) mod n = 1 mod n then N:=N,(x+y+z) mod n; fi;
  od:
 od:
od:
nops({N});
end:

a(n)=my(v=vectorsmall(n)); for(x=1,n, if(gcd(x,n)>1, next); for(y=1,x, if(gcd(y,n)>1, next); my(z=1/Mod(x*y,n)); v[lift(x+y+z)+1]=1)); sum(i=1,n, v[i]) \\ Charles R Greathouse IV, Sep 23 2024

def A376318(n):
    s = set()
    for x in range(n):
        for y in range(x,n):
            try:
                s.add((x+y+pow(x*y%n,-1,n))%n)
            except:
                continue
    return len(s) # Chai Wah Wu, Sep 23 2024

cp's OEIS Frontend

A376296 The number of solutions x<=y<=z<=w in Z/(n) of the equation x+y+z+w = x*y*z*w.

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

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Maple

Python

A376240 a(n) = number of {x,y,z} in Z/nZ such that x+y+z = x*y*z is nonzero.

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PARI

Python

A376318 The number of distinct values of x+y+z (mod n) when x*y*z = 1 (mod n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

Python

A376296 The number of solutions x<=y<=z<=w in Z/(n) of the equation x+y+z+w = xyz*w.

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

A376240 a(n) = number of {x,y,z} in Z/nZ such that x+y+z = xyz is nonzero.

A376318 The number of distinct values of x+y+z (mod n) when xyz = 1 (mod n).