Nice Sequence

Number Theory Level 4

Let $a_1,a_2,\ldots, a_n$ a sequence of integers such that $a_k$ is the number of multiples of $k$ in the sequence, for all $k = 1, 2, \ldots , n$ . How many values $n$ can take?

The answer is 2.

1 solution

Khushraj Madnani
Mar 25, 2014

Observation 1: $a i$ cannot be 0 for any i. Thus $a n$ has to be some positive number. but then there must exist a number $n$ =$a i$ for some $i\in {1,2...,n-1}$ a 1 is definitely n. Now one case is a 1=n and a n = 1. But then $a n-1$ >0 . Thus the sequence $a 2,\ldots,a n-1$ becomes non existent( not possible to build such a sequence). A formal proof can be through induction proving that for any n>2 if a 1,...a n doesn't exist then a 1,....a_n-1 will also not exst. and prove for 3 as base case.

Nice Sequence

The answer is 2.

1 solution

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