If a number can be represented as a sum of three, not necessarily distinct, powers of $2$, regrouping the equal terms (if such terms exist), one obtains a sum of at most three distinct powers of $2$, hence the base $2$ representation of such a number has at most three digits equal to $1$. Convenient numbers are those whose base $2$ representation have three digits equal to $1$, then the numbers of the form $2^a + 1 = 2^{a-1} + 2^{a-1} + 1$ with $a \in \mathbb{N}$ (numbers $2^a + 2^b$ with $a > b \ge 1$ are not convenient because $2^a + 2^b = 2^{a-1} + 2^{a-1} + 2^b = 2^{b-1} + 2^{b-1} + 2^a$); finally, the numbers $2^c = 2^{c-1} + 2^{c-1} + 2^{c-2}$ are also convenient if $c \ge 2$.

Let us count first the convenient numbers that are less than $2048$. The base $2$ representation of these numbers has at most $11$ digits. There are $C_{11}^3 = 165$ numbers less than $2048$ that can be written as a sum of three distinct powers of $2$. There are $10$ numbers of the form $2^a + 1$, $(1 \le a \le 10)$ and $9$ of the form $2^a$ $(2 \le a \le 10)$, hence $184$ numbers in total.

The numbers from $2018$ to $2047$ are larger than $2^{10} + 2^9 + 2^8$, hence their base $2$ representation has more than three digits equal to $1$, therefore none of these numbers is convenient. In conclusion, there are $184$ convenient numbers among the first $2017$.