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.