A120089 Square perimeters of primitive Pythagorean triangles.

144, 900, 3136, 8100, 17424, 23716, 33124, 43264, 54756, 57600, 93636, 115600, 139876, 144400, 166464, 174724, 207936, 213444, 244036, 298116, 304704, 357604, 414736, 422500, 476100, 490000, 541696, 571536, 640000, 722500, 746496, 756900

Offset

1,1

Comments

Square entries of A024364.

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)^2, 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),u<v are coprime with v odd}.

a(n) = A024364(k) = A000290(j) for some k and j. - R. J. Mathar, Jun 08 2006

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 : isA120089 := proc(an) RETURN( issqr(an) and isA024364(an)) ; end: for n from 2 to 1200 do if isA120089(n^2) then printf("%d, ", n^2) ; fi ; od ; # R. J. Mathar, Jun 08 2006

Mathematica

A078926[n_] := Sum[Boole[n < d^2 < 2n && CoprimeQ[d, n/d]], {d, Divisors[n/2^IntegerExponent[n, 2]]}];
Reap[For[k = 2, k <= 10^6, k += 2, If[A078926[k/2] > 0 && IntegerQ@Sqrt@k, Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Oct 25 2023 *)

Crossrefs

Cf. A120090.

Sequence in context: A234084 A223235 A187301 * A159748 A235957 A162669

Adjacent sequences: A120086 A120087 A120088 * A120090 A120091 A120092

Keyword

nonn

Author

Lekraj Beedassy, Jun 07 2006

Extensions

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

Status

approved