Maximum Possible Ratio With Digit Counting Function

Calculus Level 2

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

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

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