Using

(to avoid the extra variability from the small primes that

brings) and searching an interval around 10

^{20} I find

1 distinct prime factor: 4290

2 distinct prime factors: 21379

3 distinct prime factors: 44810

4 distinct prime factors: 54544

5 distinct prime factors: 42306

6 distinct prime factors: 22179

7 distinct prime factors: 8090

8 distinct prime factors: 2022

9 distinct prime factors: 331

10 distinct prime factors: 49

11 distinct prime factors: 1

which compares to the (naive) Landau predictions of

1 distinct prime factor: 4342

2 distinct prime factors: 16632

3 distinct prime factors: 31849

4 distinct prime factors: 40658

5 distinct prime factors: 38928

6 distinct prime factors: 29817

7 distinct prime factors: 19032

8 distinct prime factors: 10412

9 distinct prime factors: 4984

10 distinct prime factors: 2121

11 distinct prime factors: 812

and the Erdős-Kac predictions

1 distinct prime factor: 29250

2 distinct prime factors: 51228

3 distinct prime factors: 70509

4 distinct prime factors: 76319

5 distinct prime factors: 64973

6 distinct prime factors: 43492

7 distinct prime factors: 22872

8 distinct prime factors: 9438

9 distinct prime factors: 3051

10 distinct prime factors: 772

11 distinct prime factors: 152

This supports the intuition that Landau is better for small numbers of prime factors and Erdős-Kac better for large. In this case the crossover is surprisingly large (8 prime factors) but neither estimate is particularly accurate.