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 = n²
n² × n² = n⁴
n⁴ × n⁴ = n⁸
n⁸ × n⁴ = n¹²
n¹² × n² = n¹⁴
n¹⁴ × n = n¹⁵

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

n × n = n²
n² × n = n³
n³ × n³ = n⁶
n⁶ × n⁶ = n¹²
n¹² × n³ = n¹⁵

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

For 1 ≤ k ≤ 200, find ∑ m(k).