> harder for the product to be much larger than a power of 3 Unfortunately plenty of things that are hard to do still exist. Take primes for example: every tim

> harder for the product to be much larger than a power of 3 Unfortunately plenty of things that are hard to do still exist. Take primes for example: every time you pass another prime there are more primes that could be factors of larger numbers, so it must be harder and harder for large numbers to be prime. Somehow they still manage. Edit: I just quickly ran your prod(3 + 1/x) for x in {1001,1003,...,1001 + 2n} for various values of n, and empirically they are indistinguishable from uniformly distributed between powers of two on a log scale. You're going to need a better argument than "it's hard to hit a power of two with a product of (3 + 1/x)". Edit 2: (re your point 4) your floor(log2(3^n)) + 1 also seems to be an incorrect upper bound: many of the products I calculated for my first edit are above it. Edit 3: Even if that upper bound did hold, your point 5 doesn't stand up to a quick test: I followed your method for loop lengths up to 1000, and the upper bounds on x_1 bounce around a lot but tend to grow.