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A120089 Square perimeters of primitive Pythagorean triangles.

%I A120089 #15 Oct 25 2023 12:28:37
%S A120089 144,900,3136,8100,17424,23716,33124,43264,54756,57600,93636,115600,
%T A120089 139876,144400,166464,174724,207936,213444,244036,298116,304704,
%U A120089 357604,414736,422500,476100,490000,541696,571536,640000,722500,746496,756900
%N A120089 Square perimeters of primitive Pythagorean triangles.
%C A120089 Square entries of A024364.
%H A120089 Charlie Neder, <a href="/A120089/b120089.txt">Table of n, a(n) for n = 1..1000</a>
%H A120089 P. Yiu, <a href="http://math.fau.edu/yiu/RecreationalMathematics2003.pdf">"Primitive Pythagorean triangles with square perimeters" in 'Recreational Mathematics' Chap. 6.2 pp. 50/360</a>
%F A120089 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}.
%F A120089 a(n) = A024364(k) = A000290(j) for some k and j. - _R. J. Mathar_, Jun 08 2006
%p A120089 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
%t A120089 A078926[n_] := Sum[Boole[n < d^2 < 2n && CoprimeQ[d, n/d]], {d, Divisors[n/2^IntegerExponent[n, 2]]}];
%t A120089 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 *)
%Y A120089 Cf. A120090.
%K A120089 nonn
%O A120089 1,1
%A A120089 _Lekraj Beedassy_, Jun 07 2006
%E A120089 Corrected and extended by _R. J. Mathar_, Jun 08 2006