195. Let's consider the following sequence of sets of (natural) numbers. The first set $I_{0}$ consists of
two ones 1,1. Then we insert their sum $1+1=2$ between these numbers; we get the set $I_{1}: 1,2,1$. Next, between every two numbers in the set $I_{1}$, we insert the sum of these numbers; we get the set $I_{2}: 1,3,2,3,1 ;$ then, proceeding in the same way with the set $I_{2}$, we arrive at the set $I_{3}: 1,4,3,5,2,5,3,4,1$, and so on. How many times will the number 1973 appear in the millionth set $I_{1000000}$?
195. First of all, note that each of the sets $I_{0}, I_{1}$, $I_{2}, \ldots$ is obtained from the preceding one by adding a certain number of new numbers; at the same time, all the numbers that were previously present are retained in the new set. It is easy to see, furthermore, that all the new numbers appearing in the set $I_{n}$ will be greater than $n$; therefore, the number 1973 does not appear in sets with numbers greater than 1973, and all such sets contain the same number of 1973s. Let us now prove that a fixed pair of numbers $a, b$ (where $a$ stands, say, to the left of $b$; thus, $a, b$ and $b, a$ are considered different pairs!) will appear in the sequence $I_{0}, I_{1}$, $I_{2}, \ldots, I_{n}, \ldots$ of sets exactly once if the numbers $a$ and $b$ are coprime, and will not appear at all otherwise. This statement is obvious for pairs of numbers $a, b$, the largest of which does not exceed 2 (there are only two such pairs: 1,2 and 2,1 - and each of these pairs appears only once, namely, in the sequence $I_{1}: 1,2,1$); let us prove it by mathematical induction. Suppose that our statement has already been proven for all such pairs of numbers $a, b$ for which $\max [a, b] \leq 1$. "But from this it immediately follows that the pair of numbers $a, b$ appears in our sets once if $a$ and $b$ are coprime, and does not appear at all otherwise: for the numbers $a$ and $b$ are coprime if and only if the numbers $a$ and $b-\alpha$ are coprime.

Now it is clear that, since the number 1973 is prime (check this!), each of the pairs of numbers 1,1972; 2,1971; 3,1970; ...; 1971,2; 1972,1 will appear in our sets exactly once - since all these are pairs of coprime numbers. And from this it follows that the number 1973 appears in the sets $I_{n}$ with numbers $n>1973$ (in particular, in the set $I_{1}$ 000 000) exactly 1972 times (this number is equal to the number of pairs 1,1972; 2,1971; ...; 1972,1).

Note. From the solution of the problem, it also follows that any natural number $N$ appears in the sets $I_{n}$ (where $n > N$) exactly $\varphi(N)$ times, where $\varphi(N)$ is the number of numbers less than $N$ and coprime with $N$; regarding the calculation (from the number $N$) of the value $\varphi(N)$, see the Note to problem 341 (pp. 71-72).