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 A188158 Area A of the triangles such that A and the sides are integers. 57
 6, 12, 24, 30, 36, 42, 48, 54, 60, 66, 72, 84, 90, 96, 108, 114, 120, 126, 132, 144, 150, 156, 168, 180, 192, 198, 204, 210, 216, 234, 240, 252, 264, 270, 288, 294, 300, 306, 324, 330, 336, 360, 378, 384, 390, 396, 408, 420, 432, 456, 462, 468, 480, 486, 504, 510, 522, 528 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The area A of a triangle whose sides have lengths a, b, and c is given by Heron's formula: A = sqrt(s*(s-a)*(s-b)*(s-c)), where s = (a+b+c)/2. A given area often corresponds to more than one triangle; for example, a(9) = 60 for the triangles (a,b,c) = (6,25,29), (8,17,15), (13,13,10) and (13,13,24). If only primitive integer triangles (that is, the lengths of the sides are coprime) are considered, then the possible areas are 6 times the terms in A083875. - T. D. Noe, Mar 23 2011 LINKS Eric Weisstein's World of Mathematics, Triangle Wikipedia, Heronian triangle EXAMPLE a(3) = 24 because the area of the triangle whose sides are 4, 15, 13 is given by sqrt(p(p-4)(p-15)(p-13)) = 24, where p = (4 + 15 + 13)/2 = 16. MAPLE # storage of areas in T(i) T:=array(1..4000):nn:=100:k:=1:for a from 1   to nn do: for b from 1 to nn do: for c from 1 to nn do: p:=(a+b+c)/2 : x:=p*(p-a)*(p-b)*(p-c):   if x>0 then x1:=abs(x):s:=sqrt(x1) :else fi:if s=floor(s) then T[k]:=s:k:=k+1:else   fi:od:od:od: # sort of T(i) for jj from 1 to k-1 do: ii:=jj:for k1 from  ii+1 to k-1 do:if T[ii]>T[k1] then ii:=k1:else fi:od: m:=T[jj]:T[jj]:=T[ii]:T[ii]:=m:od:liste:=convert(T, set):print(liste): # second program: isA188158 := proc(A::integer)     local Asqr, s, a, b, c ;     Asqr := A^2 ;     for s in numtheory[divisors](Asqr) do         if s^2> A then         for a from 1 to s-1 do             if modp(Asqr, s-a) = 0 then                 for b from a to s-1 do                     c := 2*s-a-b ;                     if s*(s-a)*(s-b)*(s-c) = Asqr then                         return true ;                     end if;                 end do:             end if;         end do:         end if;     end do:     false ; end proc: for n from 3 to 600 do     if isA188158(n) then         printf("%d, \n", n) ;     end if; end do: # R. J. Mathar, May 02 2018 MATHEMATICA nn = 528; lst = {}; 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}]; Union[lst] (* T. D. Noe, Mar 23 2011 *) CROSSREFS Cf. A007237, A009112, A024153, A024365, A051516, A051584, A051585, A055592, A055593, A055594, A055595. Sequence in context: A074902 A096366 A247145 * A061822 A226453 A261476 Adjacent sequences:  A188155 A188156 A188157 * A188159 A188160 A188161 KEYWORD nonn AUTHOR Michel Lagneau, Mar 22 2011 STATUS approved

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Last modified February 16 14:47 EST 2019. Contains 320163 sequences. (Running on oeis4.)