hihi, all - there are 534 non-negative integers not expressable as a sum n=a^2+b^3+c^4+d^5+e^6 with a, b, c, d, e all at least 2 (i call this the strict 2 constraint), and the last few of them are 2003 0 2014 0 2051 0 2079 0 2091 0 2211 0 2366 0 2783 0 2814 0 2851 0 2854 0 2986 0 3107 0 3155 0 3302 0 3423 0 3443 0 3471 0 (pending some independent real checking; these are just what the program puts out) up to 320000, the largest count in this case is 250 for 294282 (the largest count for n <= 77317 is 113 for n=76554, and the largest count for n <= 10000 is 34 for 9905) similarly, there are 47 non-negative integers not expressable as a sum of the same form with a, b, c, d, e at least 1 (this is the strict 1 constraint), and the last of them are 108 0 113 0 115 0 202 0 206 0 232 0 256 0 656 0 up to n=10000, the largest count is 64 for 8258 in the strict 0 case, every non-negative integer can be represented, and the largest count for n <= 77317 is 268 for n=68122 (the largest count for n <= 10000 is 109 for n=9882) more soon, i think, though my program appears to have fallen into combinatoril oblivion - if it doesn't stop in a few days, i'll quit the search at 500000 (or maybe even 350000 - it's only gone 2000 in the last several hours) more later, cal