A078926 Number of primitive Pythagorean triangles with perimeter 2n.

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

Offset: 1

Author

Dean Hickerson, Dec 15 2002

Keywords

• nonn
• easy

Comments

A Pythagorean triangle is a right triangle whose edge lengths are all integers; such a triangle is 'primitive' if the lengths are relatively prime.

Equivalently, number of odd unitary divisors d of n such that sqrt(n) < d < sqrt(2n). (A divisor d of n is 'unitary' if gcd(d,n/d) = 1.) Sketch of proof: A primitive Pythagorean triangle has edge lengths (r^2-s^2, 2rs, r^2+s^2), where 1<=s

Examples

			a(858)=2; the primitive Pythagorean triangles with edge lengths (364, 627, 725) and (195, 748, 773) both have perimeter 2*858 = 1716.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..158730

Crossrefs

a(n) = A070109(2n). A078927(n) is smallest s such that a(s)=n. a(n) is nonzero iff n is in A020886.

Programs

Magma

UnitaryDivisors :=
  func;
A078926:=
  func;
[A078926(n):n in [1..105]];

Mathematica

oddpart[n_] := If[OddQ[n], n, oddpart[n/2]]; a[n_] := Length[Select[Divisors[oddpart[n]], n<#^2<2n&&GCD[ #, n/# ]==1&]]
(* Second program: *)
Table[DivisorSum[n/2^IntegerExponent[n, 2], 1 &, n < #^2 < 2 n && CoprimeQ[#, n/#] &], {n, 105}] (* Michael De Vlieger, Oct 08 2017 *)

PARI

A078926(n) = sumdiv(n,d,(d%2)*(1==gcd(d,n/d))*((d*d)>n)*((d*d)<(2*n))); \\ Antti Karttunen, Oct 07 2017

Extensions

Secondary offset added by Antti Karttunen, Oct 07 2017

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