Interesting Recreational Sequence

[Hans Havermann]

I've just added more terms and made a comment-correction to Amarnath 
Murthy's A078249 (Smallest multiple of n using all the digits other 
than used by n, or 0 if no such number exists.). The first 125 terms 
are:

203456789, 103456798, 102456789, 102357896, 123467890, 102345798, 
102345698,  102345976, 102345678, 0, 203548697, 30457896, 20465978, 
20356798, 23467890,  20357984, 20358469, 20345796, 20346587, 0, 
30457896, 103465978, 10456789,  10357896, 134678900, 10357984, 
10345698, 10347596, 10365847, 0, 20459876,  10458976, 102458697, 
10257698, 12467980, 10257948, 10254698, 10275694,  10245768, 0, 
20357689, 10358796, 10275968, 102385976, 12367890, 10327598,  10253896, 
10257936, 10263785, 0, 20348796, 10347896, 10248769, 10237698,  
123476980, 10273984, 10243698, 10234796, 10276384, 0, 20345879, 
10375948,  10245879, 10359872, 12349870, 102354978, 10253948, 10345792, 
10235874, 0,  20345689, 10345896, 10264895, 10329586, 123468900, 
10235984, 102439568,  10236954, 10246853, 0, 20346795, 10354796, 
10245769, 10253796, 12396740,  10345972, 10236594, 103254976, 10345627, 
0, 20386457, 10345768, 10247856,  10263578, 12348670, 10327584, 
10243685, 10325476, 102347586, 0, 23458967,  3457698, 2478695, 2573896, 
0, 2457398, 2365984, 2345976, 2376854, 0,  2034476598, 30479568, 
20498765, 202378956, 23468970, 23587904, 203485698,  20345796, 
20368754, 0, 30456789, 30569784, 4069578, 3597860, 346789000

The comment "a(n) = 0 if n==0 (mod 10), n uses 0 and 5 both, n uses all 
even digits, or n uses all 9 non-zero digits" needs to be corrected... 
Replace "n uses 0 and 5 both" with "n ends in 5 and uses 0". As is, 
term 501 would be 0; but it is 2396784.

The reason I'm bringing this sequence to the list is because i feel 
strongly that there ought to be more qualifiers, above and beyond those 
already stated, that will enable an n-th term not to exist. I'm 
approaching this by brute force, trying to find the smallest index-n 
for which I cannot find a(n), but some of you might be able to find 
families of such numbers by reasoning alone.

Will this sequence eventually become "all zeros" or can it be shown 
that non-zero entries exist ad infinitum?