A070109 Number of right integer triangles with perimeter n and relatively prime side lengths.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Reinhard Zumkeller, May 05 2002

Keywords

• nonn

Comments

Right integer triangles have integer areas: see A070142, A051516.

a(n) is nonzero iff n is in A024364.

Examples

			For n=30 there are A005044(30) = 19 integer triangles; only one is right: 5+12+13 = 30, 5^2+12^2 = 13^2; therefore a(30) = 1.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000 (obtained from the b-file of A078926)
• Eric Weisstein's World of Mathematics, Right Triangle.
• Eric Weisstein's World of Mathematics, Pythagorean Triples.
• Reinhard Zumkeller, Integer-sided triangles

Crossrefs

Cf. A070080, A070081, A070082, A051493, A070093, A070101, A070138, A070084, A070137.

Programs

Mathematica

unitaryDivisors[n_] := Cases[Divisors[n], d_ /; GCD[d, n/d] == 1];
A078926[n_] := Count[unitaryDivisors[n], d_ /; OddQ[d] && Sqrt[n] < d < Sqrt[2n]];
a[n_] := If[EvenQ[n], A078926[n/2], 0];
Table[a[n], {n, 1, 1716}] (* Jean-François Alcover, Oct 04 2021 *)

Formula

a(n) = A078926(n/2) if n is even; a(n)=0 if n is odd.

a(n) = A051493(n) - A070094(n) - A070102(n).

a(n) <= A024155(n).

Extensions

Secondary offset added by Antti Karttunen, Oct 07 2017