A120090 Numbers whose square is the perimeter of a primitive Pythagorean triangle.

12, 30, 56, 90, 132, 154, 182, 208, 234, 240, 306, 340, 374, 380, 408, 418, 456, 462, 494, 546, 552, 598, 644, 650, 690, 700, 736, 756, 800, 850, 864, 870, 918, 928, 986, 992, 1026, 1044, 1054, 1102, 1116, 1122, 1160, 1178, 1240, 1254, 1260, 1302, 1320

Offset

1,1

Comments

a(n) = sqrt(A120089).

Links

Charlie Neder, Table of n, a(n) for n = 1..1000

P. Yiu, "Primitive Pythagorean triangles with square perimeters" in 'Recreational Mathematics' Chap. 6.2 pp. 50/360

Formula

a(n) = 2*u*v, where u=sqrt(j/2) and v=sqrt(j+k) {for coprime pairs(j,k) j>k with odd k such that pairs (u,v) are coprime with v odd}.

Maple

isA024364 := proc(an) local r::integer, s::integer ; for r from floor((an/4)^(1/2)) to floor((an/2)^(1/2)) do for s from r-1 to 1 by -2 do if 2*r*(r+s) = an and gcd(r, s) < 2 then RETURN(true) ; fi ; if 2*r*(r+s) < an then break ; fi ; od ; od : RETURN(false) ; end : for n from 2 to 1200 do if isA024364(n^2) then printf("%d, ", n) ; fi ; od ; # R. J. Mathar, Jun 08 2006

Mathematica

isA024364[an_] := Module[{r, s}, For[ r = Floor[(an/4)^(1/2)], r <= Floor[(an/2)^(1/2)], r++, For[s = r - 1, s >= 1, s -= 2, If[2 r (r + s) == an && GCD[r, s] < 2, Return[True]]; If[2 r (r + s) < an,  Break[]]]]; Return[False]];
Select[Range[2, 2000], If[isA024364[#^2], Print[#]; True, False]&] (* Jean-François Alcover, May 24 2024, after R. J. Mathar *)

Crossrefs

Sequence in context: A131874 A111396 A080385 * A280344 A286497 A086830

Adjacent sequences: A120087 A120088 A120089 * A120091 A120092 A120093

Keyword

nonn

Author

Lekraj Beedassy, Jun 07 2006

Extensions

Corrected and extended by R. J. Mathar, Jun 08 2006

Status

approved