Following on from
12 Ways: find the smallest number that can be represented as the sum of two distinct squares of positive integers in exactly
1000 different ways.
During the time of this puzzle on the queue, Jer had made a comment that mentioned that a related sequence is http://oeis.org/A071383. Within the OEIS's discussion of that series is a link to http://www.randomwalk.de/sequences/a071383.pdf, which contains a more full series that includes A071383 as bolded entries, that extend beyond the elements listed by Sloane.
Two successive bolded entries on this list are 18643514581875625 and 25527273812106625, so they are presumably successive elements of A071383, and as far as I can tell, give the numbers that achieve new highs in the distinct number of sums of squares (specified as lattice points on a circle of given squared radius). The significance of this is that 18643514581875625 can be represented as a sum of squares in 960 ways, while 25527273812106625 can be represented in 1024 ways, so if we were looking for 1K = 1024, rather than 1000 we'd be home now.
The answer for exactly 1000 ways must be somewhat higher than 25527273812106625, as no number smaller than that can be expressed in more than 960 ways.

Posted by Charlie
on 20110923 22:16:00 