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

1,1716

Comments

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

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

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

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).

Example

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.

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 *)

Crossrefs

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

Sequence in context: A011726 A297041 A355444 * A355454 A107846 A354983

Adjacent sequences: A070106 A070107 A070108 * A070110 A070111 A070112

Keyword

nonn

Author

Reinhard Zumkeller, May 05 2002

Extensions

Secondary offset added by Antti Karttunen, Oct 07 2017

Status

approved