Number of Ones in the Binary Expansion!

Number Theory Level 5

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

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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