(In reply to
computer exploration  possible spoiler by Charlie)
UBASIC, with extended precision has verified 85 as the highest n under 10 million that satisfies the criterion.
10 sqs = 0
20 For n = 1 To 9999999
30 sqs = sqs + n * n
40 mp = Sqr(sqs * 2)
50 m = Int(mp)
60 If m * (m + 1) / 2 = sqs Then
70 :print n , Str(m) , " " , Str(sqs)
80 Next

Posted by Charlie
on 20160411 22:44:38 