A328712 Number of non-primitive Pythagorean triples with hypotenuse n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 1, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 1

Offset: 1

Author

Rui Lin, Oct 26 2019

Keywords

• nonn

Comments

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

This sequence is the count of n as hypotenuse in non-primitive Pythagorean triple.

Examples

			n=5 as hypotenuse in only one primitive Pythagorean triple, (3,4,5); so a(5)=0.
n=10 as hypotenuse in only one non-primitive Pythagorean triple, (6,8,10); so a(10)=1.
n=25 as hypotenuse in one primitive Pythagorean triple (7,24,25) and in one non-primitive Pythagorean triple (15,20,25); so a(25)=1.
		

References

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

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A046080, A024362.

Programs

Maple

f:= proc(n) local R;
if isprime(n) then return 0 fi;
  R:= map(t -> subs(t,[x,y]),[isolve(x^2+y^2=n^2)]);
  nops(select(t -> t[1]>=1 and t[2]>=t[1] and igcd(t[1],t[2])>1, R))
end proc:
map(f, [$1..100]); # Robert Israel, Oct 31 2019

Mathematica

a[n_] := Module[{R, x, y}, If[PrimeQ[n], 0, R = Solve[GCD[x, y] > 1 && x >= 1 && y >= x && x^2 + y^2 == n^2, {x, y}, Integers]; Length[R]]];
Array[a, 102] (* Jean-François Alcover, Jun 20 2020, after Maple *)

Formula

a(n) = A046080(n) - A024362(n).