# Number of Ones in the Binary Expansion!

$\Large{S_k = \sum_{n=1}^\infty \dfrac{F^k(n)}{n(n+1)} }$

Let $$F(n)$$ be the number of ones in the binary expansion of $$n$$. For Example:

• $$F(5) = F((101)_2) = 2$$
• $$F(15) = F((1111)_2) = 4$$

If $$S_1$$ can be represented as $$A\ln(B)$$ where $$A,B \in \mathbb Z^+$$ and $$B$$ isn't any perfect $$m^{th}$$ power of any integer with $$m \in \mathbb Z, \ m \geq 2$$. Find the value of $$(A-8)(B-14)$$.

Note : $$F^k(n) = (F(n))^k$$

Bonus : Generalize for $$S_k$$.

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