BALTIC-WAY 2013 Problem 16

We call a positive integer $n$ delightful if there exists an integer $k, 1<k<n$, such that

$$1+2+\cdots+(k-1)=(k+1)+(k+2)+\cdots+n$$

Does there exist a delightful number $N$ satisfying the inequalities

$$2013^{2013}<\frac{N}{2013^{2013}}<2013^{2013}+4 ?$$