How to cover all N (natural numbers) with simple poynomials of the second degree even with some overlaps.
n^2 = 1, 4, 9, 16, 25, ... n^2+1 = 2, 5, 10, 17, 26, ... n^2+2 = 3, 6, 11, 18, 27, ... etc
does cover many integers but I do not think it does work. It does not matter if there are some overlaps.
of course, in the above example, n^2+k at n=1 will eventually reach any number but let's say : can it be done non-trivially?
The problem needs to be more precise! But anyway, here's a problem which is raised by your example. Let's use n^2+k for exactly those k which are not included already: 1, 4, 9,16,25,... 2, 5,10,17,26,... 3, 6,11,18,27,... 7,10,15,22,... 8,11,16,23,... 12,15,20,27,... 13,16,21,28,... 14,17,22,29,... 19,22,27,... 24,27,... 30,... The first column 1,2,3,7,8,12,13,14,19,24,30... is not in OEIS. What is it? Brendan.