Yearly sequences

Discrete Mathematics Level 3

Define a sequence \(\displaystyle \{a_n\}_{n=1}^{n=2015}\) which satisfies the following conditions:

\[\bullet\quad \sum_{i=1}^{2015} a_{i}^{2} = 2015\\ \bullet\quad a_k\in \mathbb{Z} \setminus \{0\} \ \ \forall k\leq 2015~,k\in\mathbb{Z^+}\]

How many such sequences are possible?

The answer is of the form \(a^b\) where both \(a\) and \(b\) are non-negative integers and \(a\) is a prime number. Find the value of \((a+b)\).


\[\bullet\quad \sum_{i=1}^{n} a_i^2 = a_1^2+a_2^2+\ldots +a_{2015}^2 \ \textrm{is the sum of squares of the terms}\]

Note: This problem is an extension of a very old problem from the Brilliant community whose link I cannot seem to find. I hope you all like it.


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