Let $N(n)$ be a set of integers, which can be presented using first $n$ Egyptian fractions:

$$
N(n):=\{m\in\mathbb{Z}:\ \ m=\sum_{i=1}^n\frac{\epsilon_i}{i},\ \epsilon_i=0\ \text{or}\ 1\}
$$

I want to write a code in **Mathematica** that gives $N(n)$, but I think `A[e,n]` does not define what I need above: for example

    A[e_,n_]:=Sum[e/i,{e,0,1},{i,1,n}];
    n=1000;    
    Sum[If[IntegerQ[A[e,n]]==True,m,0],{m,1,10}]

Thanks.

P.S. example

\begin{align}
1=&1\\
2=&1+\frac12+\frac13+\frac04+\frac05+\frac16
\end{align}
therefore $N(6)=\{1,2\}$