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 2014-08-04, 14:54 #7 CRGreathouse     Aug 2006 3·1,993 Posts Using $\omega(n)$ (to avoid the extra variability from the small primes that $\Omega$ brings) and searching an interval around 1020 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.