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

Author

Rui Lin, Nov 01 2019

Keywords

• nonn

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.

Examples

			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.
		

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)

Crossrefs

Cf. A328708, A328712, A046081, A024363.

Programs

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 *)

Formula

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

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