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

Author

Lekraj Beedassy, Jun 07 2006

Keywords

• nonn

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

Programs

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 *)

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}.

Extensions

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