A376203 -id:A376203 - cp's OEIS frontend

A376202 Number of pairs 1 <= x <= y <= n-1 such that gcd(x,n) = gcd(y,n) = gcd(x+y,n) = 1 and 1/x + 1/y == 1/(x+y) mod n.

0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 18, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 36, 0, 24, 0, 0, 0, 42, 0, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 0, 0, 36, 0, 0, 0, 60, 0, 0, 0, 0, 0, 66, 0, 0, 0, 0, 0, 72, 0, 0, 0, 0, 0, 78, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 144, 0, 60, 0, 0, 0, 96, 0, 0, 0, 0, 0, 102, 0, 0

Offset: 1

Tom Duff and N. J. A. Sloane, Oct 06 2024

nonn

In general, 1/x + 1/y = 1/(x+y) is the wrong way to add fractions!
See A376203 for a(2*n-1)/2 and A376755 for a(6*n+1)/6.
From Robert Israel, Nov 06 2024: (Start)
If a(n) = 0 then a(m) = 0 whenever m is a multiple of n.
It appears that the primes p for which a(p) > 0 are A007645. (End)

			For n = 3 the a(3) = 2 solutions are (x,y) = (1,1) and (2,2).
For n = 7 the a(7) = 6 solutions are (x,y) = (1,2), (1,4), (2,4), (3,5), (3,6), (5,6).

Robert Israel, Table of n, a(n) for n = 1..6000

Cf. A000086, A007645, A046530, A290731, A376203, A376755, A376756, A376757.

a:=[];
for n from 1 to 140 do
c:=0;
for y from 1 to n-1 do
for x from 1 to y do
if gcd(y,n) = 1 and gcd(x,n) = 1 and gcd(x+y,n) = 1  and (1/x + 1/y - 1/(x+y)) mod n = 0 then c:=c+1; fi;
od: # od x
od: # od y
a:=[op(a),c];
od: # od n
a;

from math import gcd
def A376202(n):
    c = 0
    for x in range(1,n):
        if gcd(x,n) == 1:
            for y in range(x,n):
                if gcd(y,n)==gcd(z:=x+y,n)==1 and not (w:=z**2-x*y)//gcd(w,x*y*z)%n:
                    c += 1
    return c # Chai Wah Wu, Oct 06 2024

A376755 a(n) = A376202(6*n+1)/6.

Original entry on oeis.org

1, 2, 3, 0, 5, 6, 7, 7, 0, 10, 11, 12, 13, 0, 24, 16, 17, 18, 0, 0, 21, 36, 23, 0, 25, 26, 27, 26, 0, 30, 0, 32, 33, 0, 35, 60, 37, 38, 0, 40, 72, 0, 72, 0, 45, 46, 47, 0, 0, 84, 51, 52, 0, 0, 55, 56, 49, 58, 0, 57, 61, 62, 63, 0, 0, 66, 120, 68, 0, 70, 120, 72

Offset: 1

Tom Duff and N. J. A. Sloane, Oct 06 2024

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..3470

Cf. A000086, A046530, A087786, A290731, A376202, A376203, A376756, A376757.

from math import gcd
def A376755(n):
    c, m = 0, 6*n|1
    for x in range(1,m):
        if gcd(x,m) == 1:
            for y in range(x,m):
                if gcd(y,m)==gcd(z:=x+y,m)==1 and not (w:=z**2-x*y)//gcd(w,x*y*z)%m:
                    c += 1
    return c//6 # Chai Wah Wu, Oct 06 2024

a(51)-a(72) from Chai Wah Wu, Oct 06 2024

A376756 Number of pairs 0 <= x <= y <= n-1 such that x^2 + x*y + y^2 == 0 (mod n).

Original entry on oeis.org

1, 1, 3, 3, 1, 3, 7, 3, 6, 1, 1, 9, 13, 7, 3, 10, 1, 6, 19, 3, 21, 1, 1, 9, 15, 13, 18, 27, 1, 3, 31, 10, 3, 1, 7, 21, 37, 19, 39, 3, 1, 21, 43, 3, 6, 1, 1, 30, 70, 15, 3, 51, 1, 18, 1, 27, 57, 1, 1, 9, 61, 31, 60, 36, 13, 3, 67, 3, 3, 7, 1, 21, 73, 37, 45, 75, 7, 39, 79, 10, 45, 1, 1, 81, 1, 43, 3, 3, 1, 6, 163, 3, 93, 1, 19, 30, 97

Offset: 1

Tom Duff and N. J. A. Sloane, Oct 06 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000086, A046530, A087786, A290731, A376202, A376203, A376755, A376757.

def A376756(n):
    c = 0
    for x in range(n):
        z = x**2%n
        for y in range(x,n):
            if not (z+y*(x+y))%n:
                c += 1
    return c # Chai Wah Wu, Oct 06 2024

A376757 Number of pairs 0 <= x <= y <= n-1 such that x^3 == y^3 (mod n).

Original entry on oeis.org

1, 2, 3, 5, 5, 6, 13, 14, 18, 10, 11, 15, 25, 26, 15, 28, 17, 36, 37, 25, 39, 22, 23, 42, 35, 50, 81, 71, 29, 30, 61, 72, 33, 34, 65, 99, 73, 74, 75, 70, 41, 78, 85, 55, 90, 46, 47, 84, 112, 70, 51, 137, 53, 162, 55, 218, 111, 58, 59, 75, 121, 122, 288, 208, 125, 66, 133, 85, 69, 130, 71, 306, 145, 146, 105, 203, 143, 150, 157

Offset: 1

Tom Duff and N. J. A. Sloane, Oct 06 2024

nonn

A087786 includes pairs (x,y) with x>y (which are excluded from the present sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000086, A087786, A046530, A290731, A376202, A376203, A376755, A376756.

a(n) = sum(x=0, n-1, sum(y=x, n-1, Mod(x, n)^3 == Mod(y, n)^3)); \\ Michel Marcus, Oct 06 2024

from collections import Counter
def A376757(n): return sum(d*(d+1)>>1 for d in Counter(pow(x,3,n) for x in range(n)).values()) # Chai Wah Wu, Oct 06 2024

cp's OEIS Frontend

A376202 Number of pairs 1 <= x <= y <= n-1 such that gcd(x,n) = gcd(y,n) = gcd(x+y,n) = 1 and 1/x + 1/y == 1/(x+y) mod n.

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A376755 a(n) = A376202(6*n+1)/6.

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A376756 Number of pairs 0 <= x <= y <= n-1 such that x^2 + x*y + y^2 == 0 (mod n).

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A376757 Number of pairs 0 <= x <= y <= n-1 such that x^3 == y^3 (mod n).

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PARI

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