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A361416
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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.
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1
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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
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listen;
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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Table[First[
Sort@Flatten[
Table[Sqrt[x^2 + n*x*y + y^2], {x, 1, 50}, {y, x + 1, 50}]],
IntegerQ[#] &], {n, 1, 30}]
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PROG
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(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)])
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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