Integer partition equations

For some positive integers k, there exists an integer partition of the form   4^t = 2^t + k, where 4^t, 2^t, and k are all positive integers and t is a real number.

The first two such partitions are 4¹ = 2¹ + 2 and 4^1.5849625... = 2^1.5849625... + 6.

Partitions where t is also an integer are called perfect. For any m ≥ 1 let P(m) be the proportion of such partitions that are perfect with k ≤ m. Thus P(6) = 1/2.

In the following table are listed some values of P(m)

   P(5) = 1/1    P(10) = 1/2    P(15) = 2/3    P(20) = 1/2    P(25) = 1/2    P(30) = 2/5    ...    P(180) = 1/4    P(185) = 3/13

Find the smallest m for which P(m) < 1/12345

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