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

1,50

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.

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

Formula

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

Example

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.

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 *)

Crossrefs

Cf. A046080, A024362.

Sequence in context: A107015 A015374 A164058 * A319368 A092410 A100204

Adjacent sequences: A328709 A328710 A328711 * A328713 A328714 A328715

Keyword

nonn

Author

Rui Lin, Oct 26 2019

Status

approved