
COMMENTS

Also, numbers which are not the sum of "squaresminus1" (cf. A005563).  Benoit Jubin, Apr 14 2010
Conjecture: All but (n+6) positive numbers are equal to the sum of n>5 nonzero squares. For all n>5 the only (n+6) positive numbers that are not equal to the sum of n nonzero squares are {1,2,3,...,n3,n2,n1,n+1,n+2,n+4,n+5,n+7,n+10,n+13}. {2,3,5,6,7,8,10,11,12,13,14,15,17,18,19,20,21,22,23,24,26,27,28,...} = A000037 Numbers that are not squares (n=1). {1,3,4,6,7,9,11,12,14,15,16,19,21,22,23,24,27,28,30,31,33,35,36,...} = A018825 Numbers that are not the sum of 2 nonzero squares (n=2). {1,2,4,5,7,8,10,13,15,16,20,23,25,28,31,32,37,39,40,47,52,55,58,...} = A004214 Numbers that are not the sum of three nonzero squares (n=3). {1,2,3,5,6,8,9,11,14,17,24,29,32,41,56,96,128,224,384,512,...} = A000534 Numbers that are not the sum of 4 nonzero squares (n=4). {1,2,3,4,6,7,9,10,12,15,18,33} = A047701 Numbers that are not the sum of 5 nonzero squares (n=5). {1,2,3,4,5,7,8,10,11,13,16,19} Numbers that are not the sum of 6 nonzero squares (n=6). {1,2,3,4,5,6,8,9,11,12,14,17,20} Numbers that are not the sum of 7 nonzero squares (n=7). {1,2,3,4,5,6,7,9,10,12,13,15,18,21} Numbers that are not the sum of 8 nonzero squares (n=8). {1,2,3,4,5,6,7,8,10,11,13,14,16,19,22} Numbers that are not the sum of 9 nonzero squares (n=9).
The above conjecture appears as Theorem 2 on p. 73 in the Grosswald reference, where it is attributed to E. Dubouis (1911).  Wolfdieter Lang, Mar 27 2013
