According to the OEIS (A002966) there are 294314 solutions in positive integers to the equation $$\sum_{i=1}^7\frac{1}{x_i}=1$$ assuming $x_1\leq x_2\leq\cdots\leq x_7$.
Similarly for 8 summands there are 159330691 solutions.

My question: What are they? Is there a way of counting them without knowing them?

The bound for $x_n$ for $n$ summands is double exponential and I could only compute the solutions up to $n=6$ with Maple.

According to the OEIS (A002966) there are 294314 solutions in positive integers to the equation $$\sum_{i=1}^7\frac{1}{x_i}=1$$ assuming $x_1\leq x_2\leq\cdots\leq x_7$.
Similarly for 8 summands there are 159330691 solutions.

My question: What are they? Is there a way of counting them without knowing them?

The bound for $x_n$ for $n$ summands is double exponential and I could only compute the solutions up to $n=6$ with Maple.