A301384 - OEIS

%I #11 Nov 03 2019 17:04:21

%S 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 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 5,4,1,2,3,5,1,1,1,6,2,4,2,1,2,1,17,4,1

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

%C Nonzero terms of A051584.

%H Giovanni Resta, <a href="/A301384/b301384.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Triangle.html">Triangle</a>

%e 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 := proc(A::integer)

%p     local Asqr, s,a,b,c,sol ;

%p     sol := 0 ;

%p     Asqr := A^2 ;

%p     for s in numtheory[divisors](Asqr) do

%p         if s^2> A then

%p         for a from 1 to s-1 do

%p             if modp(Asqr,s-a) = 0 then

%p                 for b from a to s-1 do

%p                     c := 2*s-a-b ;

%p                     if s*(s-a)*(s-b)*(s-c) = Asqr and c >= b then

%p                         sol := sol+1 ;

%p                     end if;

%p                 end do:

%p             end if;

%p         end do:

%p         end if;

%p     end do:

%p     sol ;

%p end proc:

%p for n from 3 to 600 do

%p     a301384 := A301384(n) ;

%p     if a301384 > 0 then

%p         printf("%d,\n",a301384) ;

%p     end if;

%p end do: # _R. J. Mathar_, May 02 2018

%t 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 Cf. A188158, A024153 (distinct side lengths), A051516, A051584, A051585.

%K nonn

%O 1,2

%A _Michel Lagneau_, Mar 20 2018