A328949 Number of non-primitive Pythagorean triples with n as a leg or the hypotenuse.

0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 2, 0, 1, 3, 2, 0, 2, 0, 3, 2, 1, 0, 5, 2, 2, 2, 2, 0, 5, 0, 3, 2, 2, 3, 5, 0, 1, 3, 6, 0, 4, 0, 2, 6, 1, 0, 8, 1, 4, 3, 3, 0, 3, 3, 5, 2, 2, 0, 10, 0, 1, 5, 4, 4, 4, 0, 3, 2, 5, 0, 10, 0, 2, 7, 2, 2, 5, 0, 9, 3, 2, 0, 9, 4, 1, 3, 5, 0, 8, 3, 2, 2, 1, 3, 11, 0, 2, 5, 7

Offset

1,10

Comments

Pythagorean triples including primitive ones and non-primitive ones. For a certain n, it may be a leg or the hypotenuse in either a primitive Pythagorean triple, or a non-primitive Pythagorean triple, or both.

This sequence is the count of n as a leg or the hypotenuse in non-primitive Pythagorean triples.

References

A. Beiler, Recreations in the Theory of Numbers. New York: Dover Publications, pp. 116-117, 1966.

Links

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 5000 terms from Metin Sariyar)

Formula

a(n) = A328708(n) + A328712(n).

a(n) = A046081(n) - A024363(n).

Example

For n=10, 10 is a leg in (10,24,26) and the hypotenuse in (6,8,10), so a(10)=A328708(10)+A328712(10)=1+1=2. And 10 is not a leg or the hypotenuse in any primitive Pythagorean triple, a(10)=A046081(10)-A024363(10)=2-0=2.

Mathematica

a[n_] := Count[{x, y} /. Solve[(x^2 + y^2 == n^2 || x^2 - y^2 == n^2) && x > y > 0, {x, y}, Integers], p_ /; GCD @@ p > 1]; Array[a, 100] (* Giovanni Resta, Nov 01 2019 *)

Crossrefs

Cf. A328708, A328712, A046081, A024363.

Sequence in context: A064984 A323226 A307197 * A038555 A138108 A158777

Adjacent sequences: A328946 A328947 A328948 * A328950 A328951 A328952

Keyword

nonn

Author

Rui Lin, Nov 01 2019

Status

approved