Distinct Prime Factors

The number of distinct prime factors of a number $n$ is denoted $\omega(n)$. The first few values for $n=1$, 2, ... are 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, ... (Sloane's A001221). The first few values of the Summatory Function

$$\sum_{k=2}^n \omega(k)$$

are 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 15, 17, 19, 20, 21, ... (Sloane's A013939), and the asymptotic value is

$$\sum_{k=2}^n \omega(k)=n\ln\ln n+B_1 n+o(n),$$

where $B_1$ is Mertens Constant. In addition,

$$\sum_{k=2}^n [\omega(k)]^2=n(\ln\ln n)^2+{\mathcal O}(n\ln\ln n).$$

See also Divisor Function, Greatest Prime Factor, Hardy-Ramanujan Theorem, Heterogeneous Numbers, Least Prime Factor, Mertens Constant, Prime Factors

References

Hardy, G. H. and Wright, E. M. ``The Number of Prime Factors of $n$'' and ``The Normal Order of $\omega(n)$ and $\Omega(n)$.'' §22.10 and 22.11 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 354-358, 1979.

Sloane, N. J. A. Sequences A013939 and A001221/M0056 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.