Problem 3. Cowlendar

USACO 2024 January Contest, Silver

Bessie has woken up on a strange planet. In this planet, there are $N$ ($1\le N\le 10^4$) months, with $a_1, \ldots, a_N$ days, respectively ($1\leq a_i \leq 4 \cdot 10^9$, all $a_i$ are integers). In addition, on the planet, there are also weeks, where each week is $L$ days, with $L$ being a positive integer. Interestingly, Bessie knows the following:

• For the correct $L$, each month is at least $4$ weeks long.
• For the correct $L$, there are at most $3$ distinct values of $a_i\bmod L$.

Unfortunately, Bessie has forgotten what $L$ is! Help her by printing the sum of all possible values of $L$.

Note that the large size of integers involved in this problem may require the use of 64-bit integer data types (e.g., a "long long" in C/C++).

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains a single integer $N$. The second line contains $N$ space-separated integers, $a_1, \ldots, a_N$.

OUTPUT FORMAT (print output to the terminal / stdout):

A single integer, the sum of all possible values of $L$.

SAMPLE INPUT:

12
31 28 31 30 31 30 31 31 30 31 30 31

SAMPLE OUTPUT:

28

The possible values of $L$ are 1, 2, 3, 4, 5, 6, and 7. For example, $L=7$ is valid because each month is at least length $4 \cdot 7 = 28$ days long, and each month is either 0, 2, or 3 mod 7.

SAMPLE INPUT:

4
31 35 28 29

SAMPLE OUTPUT:

23

The possible values of $L$ are 1, 2, 3, 4, 6, and 7. For example, $L=6$ is valid because each month is at least length $4 \cdot 6 = 24$ days long, and each month is either 1, 4, or 5 mod 6.

SCORING: Inputs 3-4: $1 \leq a_i \leq 10^6$Inputs 5-14: No additional constraints

Problem credits: Brandon Wang