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

Author

Lekraj Beedassy, Jun 07 2006

Keywords

• nonn

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

Crossrefs

Cf. A120090.

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 : 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 *)

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

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

Extensions

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

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