A361416 a(n) is the least integer z for which there is a triple (x,y,z) satisfying x^2 + n*x*y + y^2 = z^2 and 0 < x < y < z.

7, 3, 11, 11, 5, 7, 11, 7, 13, 5, 9, 13, 7, 11, 25, 9, 13, 8, 11, 15, 37, 7, 17, 31, 15, 11, 25, 17, 21, 10, 19, 23, 23, 14, 25, 49, 11, 9, 73, 25, 29, 17, 27, 31, 85, 16, 21, 35, 31, 20, 49, 15, 13, 19, 35, 39, 49, 11, 41, 85, 39, 14, 47, 41, 45, 26, 19, 17

Offset

1,1

Comments

We can prove that for every positive integer n there exists a triple (x,y,z) of positive integers such that x^2 + n*x*y + y^2 = z^2. One of the solutions is (s^2 - t^2, n*t^2 + 2*s*t, s^2 + n*s*t + t^2).

Links

Table of n, a(n) for n=1..68.

Mathematics StackExchange, Finding all pairs (a,b) of positive integers such that a^2+nab+b^2 is a perfect square.

Example

a(6)=7 because there exists a triple (2,3,7) satisfying 2^2 + 6*2*3 + 3^2 = 7^2, and no smaller solution exists.

Mathematica

Table[First[
  Sort@Flatten[
    Table[Sqrt[x^2 + n*x*y + y^2], {x, 1, 50}, {y, x + 1, 50}]],
  IntegerQ[#] &], {n, 1, 30}]

Prog

(Python)
def a(n):
    for z in range(1, 100):
        for y in range(z, 1, -1):
            for x in range(1, y):
                if x**2+n*x*y+y**2==z**2:
                    return(z)
print([a(n) for n in range(1, 30)])

Crossrefs

Cf. A361417.

Sequence in context: A257324 A138859 A257328 * A076569 A061194 A248280

Adjacent sequences: A361413 A361414 A361415 * A361417 A361418 A361419

Keyword

nonn

Author

Zhining Yang, Mar 11 2023

Extensions

More terms from Sean A. Irvine, Apr 10 2023

Status

approved