$$a=\sum_{k=0}^c(k+2)^{s_k}$$ Can every positive integer eventually be expressed in this form where $s_k\ge1$ and $c\ge0$?

My Take:
First, I'll figure out the conditions for an even/odd number
Since $k^n$ is odd when odd and even when even, $(2k+1)^n\equiv1$ mod $2$, $(2k)^n\equiv0$ mod $2$.
From this, you find that for $a$ to be even, $c=4n-1,4n$. Also, for $a$ to be odd, $c=4n+1,4n+2$.

I couldn't find anything other than this, but there is a high chance it is possible to find a proof for this statement.

Notes:
$1.$ I have found 8 numbers that cannot be expressed in this form $(1,3,6,10,12,18,24,30)$, and most are multiples of 6.
$2.$ I have checked up to $400,000$ for any counterexample.