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OEIS sequence A297895 describes

Numbers that can be partitioned into squares of distinct integers whose reciprocals sum to 1.

1, 49, 200, 338, 418, 445, 486, 489, 530, 569, 609, 610, 653, 770, 775, 804, 845, 855, 898, 899, 939, 978, 1005, 1019, 1049, 1065, 1085, 1090, 1134, 1194, 1207, 1213, 1214, 1254, 1281, 1308, 1356, 1374, 1379, 1382, 1415, 1434, 1442, 1457, 1458, 1459, 1475, 1499, 1502, 1522, 1543, 1566, 1570, 1582

For example, $$ 49 = 2^2 + 3^2 + 6^2 \text{ and } \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = 1;\\ 200 = 2^2 + 4^2 + 6^2 + 12^2 \text{ and } \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{12} = 1. $$


The author claims that

All integers $\geq 8543$ belong to this sequence.

What is a proof (or even a heuristic) that explains why this is the case?

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    $\begingroup$ I sent an email to the author (Max Alekseyev) and he said: "I have just submitted a paper proving exactly this. I'll post a preprint to arXiv some time this week. So stay tuned." $\endgroup$ – Peter Kagey Jan 16 '18 at 22:32
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As promised, here is my preprint with a proof that all integers $\geq 8543$ belong to A297895:

On partitions into squares of distinct integers whose reciprocals sum to 1

Questions and comments are welcome.

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