Hanoi Open Mathematics Competition 2012 Senior Question 6

For \(n=2,3,\ldots\), put \(A_n=\left(1-\dfrac1{1+2}\right)\left(1-\dfrac1{1+2+3}\right)\cdots\left(1-\dfrac1{1+2+\cdots+n}\right)\). Determine all integers \(n\ge2\) for which \(1/A_n\) is an integer.