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

1,858

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<r, r and s are relatively prime and r+s is odd. This has perimeter 2n iff n=r(r+s). Let d=r+s.

Links

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

Example

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

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 *)

Prog

(Magma) UnitaryDivisors :=
  func<n| [d:d in Divisors(n)|GCD(d, n div d) eq 1]>;
A078926:=
  func<n | #[d:d in UnitaryDivisors(n)| IsOdd(d) and Isqrt(n) lt d and d le Isqrt(2*n-1)] >;
[A078926(n):n in [1..105]];
(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

Crossrefs

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

Sequence in context: A378537 A302049 A373383 * A324824 A025458 A378449

Adjacent sequences: A078923 A078924 A078925 * A078927 A078928 A078929

Keyword

nonn,easy

Author

Dean Hickerson, Dec 15 2002

Extensions

Secondary offset added by Antti Karttunen, Oct 07 2017

Status

approved