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A301384 Number of integer-sided triangles of area A188158(n).

%I A301384 #12 Feb 16 2025 08:33:53
%S A301384 1,2,2,1,2,1,2,1,4,1,1,4,2,2,2,1,4,3,1,2,1,2,4,2,2,1,1,6,3,1,5,3,3,2,
%T A301384 2,1,4,1,2,4,8,5,1,2,1,3,1,15,2,4,2,1,5,1,6,2,1,3,4,3,1,2,2,1,2,4,5,1,
%U A301384 5,4,1,2,3,5,1,1,1,6,2,4,2,1,2,1,17,4,1
%N A301384 Number of integer-sided triangles of area A188158(n).
%C A301384 Nonzero terms of A051584.
%H A301384 Giovanni Resta, <a href="/A301384/b301384.txt">Table of n, a(n) for n = 1..10000</a>
%H A301384 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Triangle.html">Triangle</a>
%e A301384 a(9) = 4 because A188158(9) = 60 corresponding to 4 triangles (a, b, c) = (6, 25, 29), (8, 15, 17), (10, 13, 13) and (13, 13, 24) of area 60.
%p A301384 A301384 := proc(A::integer)
%p A301384     local Asqr, s,a,b,c,sol ;
%p A301384     sol := 0 ;
%p A301384     Asqr := A^2 ;
%p A301384     for s in numtheory[divisors](Asqr) do
%p A301384         if s^2> A then
%p A301384         for a from 1 to s-1 do
%p A301384             if modp(Asqr,s-a) = 0 then
%p A301384                 for b from a to s-1 do
%p A301384                     c := 2*s-a-b ;
%p A301384                     if s*(s-a)*(s-b)*(s-c) = Asqr and c >= b then
%p A301384                         sol := sol+1 ;
%p A301384                     end if;
%p A301384                 end do:
%p A301384             end if;
%p A301384         end do:
%p A301384         end if;
%p A301384     end do:
%p A301384     sol ;
%p A301384 end proc:
%p A301384 for n from 3 to 600 do
%p A301384     a301384 := A301384(n) ;
%p A301384     if a301384 > 0 then
%p A301384         printf("%d,\n",a301384) ;
%p A301384     end if;
%p A301384 end do: # _R. J. Mathar_, May 02 2018
%t A301384 nn=1000;lst={};lst2={};Do[s=(a+b+c)/2;If[IntegerQ[s],area2=s(s-a)(s-b)(s-c);If[0<area2<=nn^2&&IntegerQ[Sqrt[area2]],AppendTo[lst,Sqrt[area2]]]],{a,nn},{b,a},{c,b}];lst1=Sort[lst];n0=Length[lst];i=1;Do[If[Part[lst1,t-1]==Part[lst1,t],i=i+1,AppendTo[lst2,i];i=1],{t,2,n0}];lst2
%Y A301384 Cf. A188158, A024153 (distinct side lengths), A051516, A051584, A051585.
%K A301384 nonn
%O A301384 1,2
%A A301384 _Michel Lagneau_, Mar 20 2018