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!