A078927 -id:A078927 - cp's OEIS frontend

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

Dean Hickerson, Dec 15 2002

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

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

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

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

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

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 *)

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

Secondary offset added by Antti Karttunen, Oct 07 2017

A078928 Smallest p for which there are exactly n primitive Pythagorean triangles with perimeter p; i.e., smallest p such that A070109(p) = n.

Original entry on oeis.org

12, 1716, 14280, 317460, 1542684, 6240360, 19399380, 63303240, 239168580, 397687290, 458948490, 813632820, 562582020, 2824441620, 3346393050, 6915878970, 6469693230, 8720021310, 9146807670, 8254436190, 23065862820, 25859373540, 202536455550

Offset: 1

Dean Hickerson, Dec 15 2002

nonn

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

Least perimeter common to exactly n primitive Pythagorean triangles. - Lekraj Beedassy, May 14 2004

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

Derek J. C. Radden and Peter T. C. Radden, Table of n, a(n) for n=1..39 (terms 1 through 15 were computed by Derek J. C. Radden)
Shyam Sunder Gupta, Number Curiosities, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 23, 567-604.
C. B. T. (Reviewer), Review of Andrew S. Anema, A table of primitive Pythagorean triangle with identical perimeters, Mathematical Tables and Other Aids to Computation, Vol. 10, No. 53 (Jan., 1956), pp. 35-36.

a(n) = 2*A078927(n). Cf. A070109.

oddpart[n_] := If[OddQ[n], n, oddpart[n/2]]; ct[p_] := Length[Select[Divisors[oddpart[p/2]], p/2<#^2

a(8) from Robert G. Wilson v, Dec 19 2002
a(9)-a(15) from Derek J C Radden, Dec 22 2012
a(16)-a(39) from Peter T. C. Radden, Dec 29 2012

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A078926 Number of primitive Pythagorean triangles with perimeter 2n.

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