*Factors*

Define $\mathbf{[n]} := \{1, 2, 3, \ldots, n\}.$ Then

• $\bigg|\{x \in \mathbf{[2017]} : x \text{ has one positive factor}\}\bigg| = \bigg|\{1\}\bigg| = 1$
• $\bigg|\{x \in \mathbf{[2017]} : x \text{ has two positive factors}\}\bigg| = \bigg|\{x \in \mathbf{[2017]} : x \text{ is prime}\}\bigg| = 306$
• Since $\sqrt{2017} \approx 44.9,$ $\bigg|\{x \in \mathbf{[2017]} : x \text{ has three positive factors}\}\bigg| = \bigg|\{x \in \mathbf{[2017]} : x=p^2 \text{ for some prime } p\}\bigg| = \bigg|\{x \in \mathbf{[44]} : x \text{ is prime}\}\bigg| = 14$

Therefore the value we're trying to find is $\bigg|\{x \in \mathbf{[2017]} : x \text{ has at least four positive factors}\}\bigg| = 2017 - 1 - 306 - 14 = \boxed{1696}$