

A218243


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


2



60, 150, 200, 480, 780, 1200, 1530, 1600, 1620, 1690, 1950, 2040, 2100, 2730, 2860, 3570, 3840, 4050, 4056, 4200, 4350, 4624, 5100, 5400, 5460, 6240, 7500, 8120, 8250, 8670, 8750, 9600, 10812, 11050, 11900, 12180, 12240, 12800, 12960, 13260, 13520, 13650
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OFFSET

1,1


COMMENTS

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*(sa)*(sb)*(sc)), where s = (a+b+c)/2 => A = sqrt(6(63)(64)(65)) = 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(164)(1613)(1615))=24 and the area of the triangle (5,12,13) is sqrt(15(155)(1512)(1513))=30.
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.
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).
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
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


LINKS

Eric Weisstein's World of Mathematics, Triangle


EXAMPLE

60 is in the sequence because 60 = 3*4*5 and the corresponding area is sqrt(6(63)(64)(65)) = 6 = A188158(1).


MATHEMATICA

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 *)


PROG

(PARI) Heron(a, b, c)=a*=a; b*=b; c*=c; ((a+b+c)^22*(a^2+b^2+c^2))
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


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



