Problem of the Week [June 17, 2013]

Suppose  is a positive integer. Consider an upward-pointing equilateral triangle with side length , cut up into unit triangles. For each , let    represent the total number of downward-pointing equilateral triangles of all sizes. For example,   and . Determine, with justification, all positive integers  for which  is divisible by .

Correct answers as well as solutions will be acknowledged.