Let

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

I want to write a code in Mathematica as follows 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.