A sequence of natural numbers $(a_n)$ with the property that all pairwise sums of elements are distinct is called a *Sidon sequence* and it is proved there are at most $s(n)\sim\sqrt n$ elements of such a sequence below $n$.

My question is:

Is there an (infinite) Sidon sequence with the property:

There is an $M>0$ such that for every natural $n\geq M$ there exist $i,j$ with $n=a_i+a_j$.

(We include the case where $i=j$ )

I call this kind of sequence "complete" since it covers all positive integers from some point on.

I guess that such a sequence does not exist and surely we must have $M>1$ (this easy to prove)

I was not able to find a proof or reference.

Any help would be appreciated!