A Curious List

Number Theory Level 4

I write a list beginning with the number $1$ . Then, I write all the multiples of $2$ until ${ 2 }^{ 2 }$ . Next, I write all the multiples of $3$ until ${ 3 }^{ 2 }$ , and I continue writing numbers successively. The first numbers of the list are $1, 2, 4, 3, 6, 9, 4, 8, 12, 16, 5,\ldots$

Find the sum of the digits of the position number in which 2016 appears for the first time.