Run time bounds for a model related to sieving

The following problem is very close to one arising in
factorization algorithms.

       Generate random numbers in the interval [1,N]
       until some subset has a product which is a square.
       How long does this take (and how can you tell when
       you're done)?

Crude analogies suggest that this stopping time has a very
sharp threshold.  We prove upper and lower bounds that are
within a factor of 4/pi and conjecture that our upper bound
is asymptotically sharp.

Joint work with Andrew Granville, Ernie Croot and Prasad Tetali.