New Year's Countdown Day 15: Anti-magical Diagonals

Define a heterosquare to be a square grid containing consecutive positive integers starting from 1 such that the sums of the integers in each row, column, and long diagonal are all different. Furthermore, define an anti-magic square to be a heterosquare whose sums form a consecutive sequence of integers.

If there exists a $15 \times 15$ anti-magic square, then there are $n$ possible values $d_1, d_2, \dots, d_n$ for the sum of its two diagonal sums. Find the value of $n \displaystyle \sum_{i = 1}^n d_i.$

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Note: A diagonal sum is the sum of the numbers along a long diagonal of the square. For example, the two diagonal sums of the following square are $1 + 5 + 9 = 15$ and $3 + 5 + 7 = 15:$ $\begin{array}{|c|c|c|} \hline 1 & 2 & 3 \\ \hline 4 & 5 & 6 \\ \hline 7 & 8 & 9 \\ \hline \end{array}$

• This problem is related to Open Problem #2 of the Brilliant.org Open Problems Group.

• This problem is part of the set New Year's Countdown 2017.