An equivalent form of that representation is $2^k(1 + 2^{\ell} + 2^{2\ell} + \cdots + 2^{m\ell})$, based on the fact that $2^p-1 \mid 2^q-1$ if and only if $p\mid q$. The first not representable is $11 = 1+2+2^3$ (since $1=1$, $3=1+2^1$, $5=1+2^2$, $7=1+2^1+2^2$, $9=1+2^3$).

Appeared in All-Russian 2005, 10.1

Basically, a number is representable in this form iff the positions of the $1$s in its binary representation are in arithmetic progression.

The number $n$ cannot be represented in this form if it's binary sum i.e $n=\sum2^{a_i}$, where all $a_i$ are distinct,has exponents $a_i$ and $a_i$ are not in arithmetic progression. We see the first such no. is $11=2^3+2^2+2^0$, where $3,2,0$ are not in A.P.