Problem 122: Efficient exponentiation

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The most naive way of computing n15 requires fourteen multiplications:

n × n × ... × n = n15

But using a "binary" method you can compute it in six multiplications:

n × n = n2n2 × n2 = n4n4 × n4 = n8n8 × n4 = n12n12 × n2 = n14n14 × n = n15

However it is yet possible to compute it in only five multiplications:

n × n = n2n2 × n = n3n3 × n3 = n6n6 × n6 = n12n12 × n3 = n15

We shall define m(k) to be the minimum number of multiplications to compute nk; for example m(15) = 5.

For 1 ≤ k ≤ 200, find ∑ m(k).