Why does this sequence exist?

[David Wasserman]

I just sent this to Neil.  I had a lot of fun with it.  It took me a while to figure out why the sequence never hits a dead end.  You can read my proof, or figure it out yourself.

%I A082218
%S A082218 1,3,5,6,7,2,10,12,8,4,9,14,13,16,19,25,15,37,21,23,11,20,17,22,29,26,
%T A082218 35,24,28,36,18,32,38,44,40,48,31,56,33,68,43,50,39,34,41,27,47,61,53,
%U A082218 57,45,75,85,93,55,30,49,65,63,72,67,88,69,62,73,51,81,83,80,70,128,42
%N A082218 Square array in which for every k, the kth partial sums of every row and column are divisible by k.  Array r\
  ead by antidiagonals, alternating upwards and downwards.  Each entry is the least number not already used that fits t\
  he divisibility requirement.
%C A082218  T(i, j) must satisfy a congruence mod i, and another congruence mod j.  i and j are not always relatively pr\
  ime, but this pair of congruences is always solvable.  See the link for a proof. - David Wasserman
%H A039508 D. Wasserman, <a href="http://home.earthlink.net/~dwasserm/A082218.html">Proof</a>
%e A082218  1  3  2 10 19 25...
%e A082218  5  7 12 16 15...
%e A082218  6  8 13 37...
%e A082218  4 14 21...
%e A082218  9 23...
%e A082218 11...
%e A082218 ...
%Y A082218 Cf. A082219, A082220, A082221, A082222, A082223.
%K A082218 nonn,easy,tabf
%O A082218 1,2
%A A082218 Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 09 2003
%E A082218 Edited and extended by David Wasserman (dwasserm(AT)earthlink.net), Aug 26 2004.