Tougher Throwing Numbers

Consider the sequence of positive integers \(1,2,3,4,5,6, \ldots \). An operation throw consists of taking the first number and move it to the back of the \((k+1)^\text{th}\) number where \(k\) is the second number in the sequence. For example,

1
2
3
4
5
6
throw 0 : 1 2 3 4 5 6 7 8
throw 1 : 2 3 1 4 5 6 7 8
throw 2 : 3 1 4 2 5 6 7 8
throw 3 : 1 3 4 2 5 6 7 8
throw 4 : 3 4 2 1 5 6 7 8
throw 5 : 4 2 1 5 3 6 7 8

The first occurrence of 2, 3, 4, 5 being the head of the sequence are after 1, 2, 5 and 12 throws, respectively. How many throw operations do you need to perform to get 20 in the head of the sequence?


You are encouraged to solve an easier version of this problem first.
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