Competition problem — Number theory

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Number theory Difficulty 5.2 AIME, harder Prove it Belarus

A sequence is called N-nice if it consists of a finite number of the consecutive positive integers and the sum of all its terms is equal to $N$ .
Find the total number of all $4030^k$ -nice sequences, where $k$ is a positive integer. (N-nice sequence may consist of a single term.)

Solution

Let $P(N)$ be the number of $N$ -nice sequences, $D(N)$ be the number of odd divisors of $N$ . Then $P(N) = D(N)$ (see the solution of Problem C.7). Since $D(4030^k) = D(2^k \cdot 5^k \cdot 13^k \cdot 31^k) = (k+1)^3$ , we see that the number of the required $4030^k$ -nice sequences is equal to $(k+1)^3$ .

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