Question 27
Let $t(n)$ denote the sum of the digits in the binary representation of a positive integer $n$, and let $k \ge 2$ be an integer.
a. Show that there exists a sequence $(a_i)_{i=1}^\infty$ of integers such that $a_m \ge 3$ is an odd integer and $t(a_1 a_2 \cdots a_m) = k$ for all $m \ge 1$.
b. Show that there is an integer $N$ such that $t(3 \cdot 5 \cdots (2m + 1)) \gt k$ for all integers $m \ge N$.
a. Show that there exists a sequence $(a_i)_{i=1}^\infty$ of integers such that $a_m \ge 3$ is an odd integer and $t(a_1 a_2 \cdots a_m) = k$ for all $m \ge 1$.
b. Show that there is an integer $N$ such that $t(3 \cdot 5 \cdots (2m + 1)) \gt k$ for all integers $m \ge N$.