5. The number of non-empty subsets of the set $\{1,2, \cdots, 2016\}$ whose elements sum to an odd number is $\qquad$
5. $2^{2015}$
$$
\text { Let } f(x)=(1+x)\left(1+x^{2}\right) \cdots\left(1+x^{2016}\right) \text {. }
$$

Then the desired value is the sum of the coefficients of the odd powers of $x$ in the expansion of $f(x)$. Therefore, the number of non-empty subsets with an odd sum of elements is $\frac{1}{2}(f(1)-f(-1))=2^{2015}$.