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x_n/ y_n =\sum_{k=1}^{n}{\frac{1}{k {n \choose k}}}, y_n not divisible by 2^n

Source: Mathley 2014.1 p8

August 20, 2020

number theorydivisiblepower of 2divides

Problem Statement

For every

n

positive integers we denote

\frac{x_n}{y_n}=\sum_{k=1}^{n}{\frac{1}{k {n \choose k}}}

where

x_n, y_n

are coprime positive integers. Prove that

y_n

is not divisible by

2^n

for any positive integers

n

.
Ha Duy Hung, high school specializing in the Ha University of Education, Hanoi, Xuan Thuy, Cau Giay, Hanoi