Is it possible to check if a number is the product of two primes without factorizing it? I have a large number which I suspect may be a private RSA key (although its size, at 613 bits, seems a bit unorthodox). 

I have started to run a factorization algorithm on it, and after a few hours it still has not found any factors. But I wouldn't want to keep the algorithm running for a couple more days (or pay for a few EC2 instances to run the factorization on) only to find out that it is simply the product of three or more somewhat large prime numbers. 

So, I was wondering if there was a way (even theoretical!) to check if that number was indeed the product of two prime numbers without actually having to factorize it.
Given a 613-bit integer, it is easy to check that it is composite (and we assume that's been done). It can be hard to factor, but that's feasible with GNFS, including with ready-made tools (see Luke Valenta, Shaanan Cohney, Alex Liao, Joshua Fried, Satya Bodduluri, Nadia Heninger's Factoring as a Service), and that's one sure way to demonstrate how many factors it has.

We can do better! We can use Lenstra's ECM repeatedly, with parameters and runtime tuned to find factors less than about 205 bits. That's within reach of the best current implementations, see Paul Zimmermann's list of ECM factoring records. If that fails, then it is a very strong indication that there is no factor 205-bit or less, thus there must be only two factors (for the product of three or more 205-bit integers has at least 613 bits). If that succeeds, we can test the primality and/or factor the rest.

I believe the tool of choice is GMP-ECM.

Edit: That is not generalizable to much larger integers. Starting about 800 bits, it becomes quite hard to tell apart those with two factors, and those with three factors of about equal size.