Problem

Source: Turkish TST 2011 Problem 6

Tags: floor function, limit, number theory, greatest common divisor, induction, prime factorization, number theory proposed



Let $t(n)$ be the sum of the digits in the binary representation of a positive integer $n,$ and let $k \geq 2$ be an integer. a. Show that there exists a sequence $(a_i)_{i=1}^{\infty}$ of integers such that $a_m \geq 3$ is an odd integer and $t(a_1a_2 \cdots a_m)=k$ for all $m \geq 1.$ b. Show that there is an integer $N$ such that $t(3 \cdot 5 \cdots (2m+1))>k$ for all integers $m \geq N.$