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A132404 Smallest short legs 'A' of exactly n primitive Pythagorean triangles.

%I A132404 #17 Oct 25 2015 17:34:20
%S A132404 3,20,60,204,5040,420,660,2040
%N A132404 Smallest short legs 'A' of exactly n primitive Pythagorean triangles.
%C A132404 Where records occur in A024359. a(12) = 13860, a(13) = 4620, and a(14) = 7140. - _T. D. Noe_, Feb 23 2012
%C A132404 From _Colin Barker_, Oct 25 2015: (Start)
%C A132404 a(11) = 872100, a(15) = 22440 and a(16) = 55440.
%C A132404 a(9), a(10), a(17), a(18), a(19) and a(20) are not less than 6000000.
%C A132404 (End)
%e A132404 The solutions for the first 7 are
%e A132404 1, (3,4,5)
%e A132404 2, (20,21,29), (20,99,101)
%e A132404 3, (60,91,109), (60,221,229), (60,899,901)
%e A132404 4, (204,253,325), (204,1147,1165), (204,2597,2605), (204,10403,10405)
%e A132404 5, (5040,78319,78481), (5040,99161,99289), (5040,129551,129649), (5040,253991,254041), (5040,6350399,6350401)
%e A132404 6, (420,851,949), (420,1189,1261), (420,1739,1789), (420,4891,4909), (420,11021,11029), (420,44099,44101)
%e A132404 7, (660,779,1021), (660,989,1189), (660,2989,3061), (660,4331,4381), (660,12091,12109), (660,27221,27229), (660,108899,108901)
%t A132404 PyphagoreanAs[a_] := (q={}; k=0; If[a>=8,r=4,r=1]; Do[y=(a^2+b^2)^0.5; c=IntegerPart[y]; If[c==y, p=0; If[GCD[a,b,c]==1, AppendTo[q,a.b.c]; k++ ]], {b,a+1,a^2/r}]; PrependTo[q,k]; q); lst={}; x=0; Do[w=PyphagoreanAs[n][[1]]; If[w>x, Print[Date[],"A=",n,",w=",w]; AppendTo[lst,n]; x=w], {n,1000}]; lst
%t A132404 solns[a_] := Module[{b, c, soln}, soln = Reduce[a^2 + b^2 == c^2 && a < b && c > 0 && GCD[a, b, c] == 1, {b, c}, Integers]; If[soln === False, 0, If[soln[[1, 1]] === b, 1, Length[soln]]]]; nn = 20; t = Table[0, {nn}]; Do[s = solns[n]; If[s > nn, Print[{s, n}], If[t[[s]] == 0, t[[s]] = n; Print[{s, n}]]], {n, 5040}]; t (* _T. D. Noe_, Feb 23 2012 *)
%Y A132404 Cf. A024359.
%K A132404 nonn,more
%O A132404 1,1
%A A132404 _Vladimir Joseph Stephan Orlovsky_, Aug 26 2008
%E A132404 a(5) from _T. D. Noe_, Feb 23 2012