Maximum Possible Ratio With Digit Counting Function

Calculus Level 3

Let \(d(n)\) denote the number of digits of \(n\). For example, \(d(1929) = 4\) and \(d(301) = 3\).

What is the maximum value of the ratio \[\dfrac{\big(d(n)\big)^6}{n}\] as \(n\) ranges over the positive integers?

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