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A218243 Triangle numbers: m = a*b*c such that the integers a,b,c are the sides of a triangle with integer area.

%I A218243 #23 Feb 19 2025 05:44:29
%S A218243 60,150,200,480,780,1200,1530,1600,1620,1690,1950,2040,2100,2730,2860,
%T A218243 3570,3840,4050,4056,4200,4350,4624,5100,5400,5460,6240,7500,8120,
%U A218243 8250,8670,8750,9600,10812,11050,11900,12180,12240,12800,12960,13260,13520,13650
%N A218243 Triangle numbers: m = a*b*c such that the integers a,b,c are the sides of a triangle with integer area.
%C A218243 A triangle number m is an integer with at least one decomposition m = a*b*c such that the area of the triangle of sides (a,b,c) is an integer. Because this property is not always unique, we introduce the notion of "triangle order" for each triangle number m, denoted by TO(m). For example, TO(60) = 1 because the decomposition 60 = 3*4*5 is unique with the triangle (3,4,5) whose area A is given by Heron's formula: A = sqrt(s*(s-a)*(s-b)*(s-c)), where s = (a+b+c)/2 => A = sqrt(6*(6-3)*(6-4)*(6-5)) = 6, but TO(780) = 2 because 780 = 4*13*15 = 5*12*13 and the area of the triangle (4,13,15) is sqrt(16*(16-4)*(16-13)*(16-15)) = 24 and the area of the triangle (5,12,13) is sqrt(15*(15-5)*(15-12)*(15-13)) = 30.
%C A218243 Given an area A of A188158, there exists either a unique triangle number (for example for A = 6 => m = 60 = 3*4*5), or several triangle numbers (for example for A=60 => m1 = 4350 = 6*25*29, m2 = 2040 = 8*15*17, m3 = 1690 = 13*13*10).
%C A218243 The number of ways to write m = a*b*c with 1<=a<=b<=c<=m is given by A034836, thus: TO(m) <= A034836(m).
%C A218243 If n is in this sequence, so is nk^3 for any k > 0. Thus this sequence is infinite. - _Charles R Greathouse IV_, Oct 24 2012
%C A218243 In view of the preceding comment, one might call "primitive" the elements of the sequence for which there is no k>1 such that n/k^3 is again a term of the sequence. These elements 60, 150, 200, 780, 1530, 1690, 1950,... are listed in A218392. - _M. F. Hasler_, Oct 27 2012
%H A218243 Charles R Greathouse IV, <a href="/A218243/b218243.txt">Table of n, a(n) for n = 1..2000</a>
%H A218243 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Triangle.html">Triangle</a>
%e A218243 60 is in the sequence because 60 = 3*4*5 and the corresponding area is sqrt(6*(6-3)*(6-4)*(6-5)) = 6 = A188158(1).
%t A218243 nn = 500; 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, a*b*c]]], {a, nn}, {b, a}, {c, b}]; Union[lst] (* Program from _T. D. Noe_, adapted for this sequence - see A188158 *)
%o A218243 (PARI) Heron(a,b,c)=a*=a;b*=b;c*=c;((a+b+c)^2-2*(a^2+b^2+c^2))
%o A218243 is(n)=fordiv(n,a, if(a^3<=n, next); fordiv(n/a,b, my(c=n/a/b,h); if(a>=b && b>=c && a<b+c, h=Heron(a,b,c); if(h%16==0 && issquare(h), return(1)))));0 \\ _Charles R Greathouse IV_, Oct 24 2012
%Y A218243 Subsequence of A139270.
%Y A218243 Cf. A034836, A188158, A083875.
%K A218243 nonn
%O A218243 1,1
%A A218243 _Michel Lagneau_, Oct 24 2012