Find all integers 1<=k<=169 for which 169 is not the sum of k nonzero squares.

The squares are not necessarily unique. For example k=5: 169=1+4+4+16+144.

This should be done without a brute force program.

k=1   1*169

k=2   1*144 + 1*25

k=3   1*144 + 1*16 + 1*9

k=4   1*100 + 1*49 + 1*16 + 1*4

k=5   2*64 + 1*36 + 1*4 + 1*1

k=6   2*81 + 1*4 + 3*1

k=7   2*64 + 1*25 + 4*4

k=8   2*64 + 1*36 + 5*1

k=9   2*81 + 7*1

k=10  2*64 + 1*25 + 3*4 + 4*1

k=11  2*64 + 1*16 + 1*9 + 3*4 + 4*1

With 100 = 64 + 36, 25 = 16 + 9, 9 = 4 + 4 + 1 it looks as though I should be able to get all values of k up to 149 (see previous entry).  Leaving 150, 153, 156, 159, 162, 164, 165, 167 & 168 as values of k I can't find.

Posted by Nosher on 2004-11-04 22:59:44