Question 136
Consider the infinite sequences of positive integer numbers, in which each positive integer number to meet among elements of this sequence is equal once. Let $\{a_n\}$, $n \ge 1$ be such a sequence. Name it sequence "consecutive" if for each positive integer number $k$ and for any positive integer numbers $n$ and $m$ such that $a_n \lt a_m$, the following inequality holds: $a_{kn} \lt a_{km}$. For example, the sequence $a_n = n$ is "consecutive".
a) Prove that there exists a "consecutive" sequence which is different from $a_n = n$.
b) Does there exist a "consecutive" sequence for which the inequality $a_n \neq n$, $n \geq 2$ holds?
c) Does there exist a "consecutive" sequence for which the inequality $a_n \neq n$, $n \geq 1$ holds?
a) Prove that there exists a "consecutive" sequence which is different from $a_n = n$.
b) Does there exist a "consecutive" sequence for which the inequality $a_n \neq n$, $n \geq 2$ holds?
c) Does there exist a "consecutive" sequence for which the inequality $a_n \neq n$, $n \geq 1$ holds?