smc

If the second size has length x, then the first side has length 3x, and we have the third side which has length 15. By the triangle inequality, we have: $$x+15>3x\Rightarrow 2x<15\Rightarrow x<7.5$$ Now, since we want the greatest perimeter, we want the greatest integer x, and if $x<7.5$ then $x=7$. Then, the first side has length $3\ast 7=21$, the second side has length $7$, the third side has length $15$, and so the perimeter is $21+7+15=43\Rightarrow \boxed{43}$.