A228847 Let A(x,y,z) denote the integer area A of the triangle with integer side lengths (x,y,z). a(n) is the smallest area such that there exist at least n distinct triangles (x_i, y_i, z_i) satisfying A(x_1, y_1, z_1) = A(x_2, y_2, z_2) = ... = A(x_n, y_n, z_n).

6, 12, 60, 60, 210, 210, 336, 336, 420, 420, 420, 420, 420, 420, 420, 840, 840, 1680, 1680, 1680, 1680, 1680, 1680, 7560, 7560, 7560, 7560, 7560, 7560, 7560, 7560, 7560, 7560, 7560, 7560, 7560, 7560, 7560, 7560, 7560, 7560, 7560, 7560, 7560, 7560, 7560, 7560

Offset: 1

Author

Michel Lagneau, Sep 05 2013

Keywords

• nonn

Comments

The area A of a triangle whose sides have lengths x, y, and z is given by A188158. The area is given by Heron's formula: A = sqrt(s*(s-a)*(s-b)*(s-c)), where s = (a+b+c)/2.

The sequence A228858 gives the areas such that there exist exactly n triangles (x_i, y_i, z_i) satisfying A(x_1, y_1, z_1) = A(x_2, y_2, z_2) = ... = A(x_n, y_n, z_n).

The Maple program examines all triangles having longest side <= 600.

Examples

			a(6) = 210 because there exist 6 triangles:
(3,148,149) => A = sqrt(150*(150-3)*(150-148)*(150-149)) = 210;
(7,65,68) => A = sqrt(70*(70-7)*(70-65)*(70-68)) = 210;
(12,35,37) => A = sqrt(42*(42-12)*(42-35)*(42-37)) = 210;
(17,25,28) => A = sqrt(35*(35-17)*(35-25)*(35-28)) = 210;
(17,28,39) => A = sqrt(42*(42-17)*(42-28)*(42-39)) = 210;
(20,21,29) => A = sqrt(35*(35-20)*(35-21)*(35-29)) = 210.
		

Links

• Eric Weisstein's World of Mathematics, Triangle

Crossrefs

Cf. A188158, A228858.

Programs

Maple

with(numtheory):nn:=600:lst:={}:T:=array(1..30000):k:=0:for a from 1 to nn do: for b from a to nn  do: for c from b 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 lst:=lst union {s}:k:=k+1:T[k]:=s:else fi:od:od:od:k1:=nops(lst):for n from 1 to 30 do:jj:=0:for i from 1 to k1 while(jj=0) do:ii:=0:for j from 1 to k while(jj=0) do:if lst[i]=T[j] then ii:=ii+1:if ii=n then jj:=1:printf ( "%d %d \n",n,lst[i]):else fi:fi:od:od:od: