If I count in binary or any system using the 0 very few patterns can be seen relating to the primes,complexity, The Triangle numbers in all dimensions or The factorial numbers.

When I start to count with out the 0 instead of with 1s and 0s with 1s and 2s things get interesting.
example.
Set 1..
1(+-s*)(1), 2(s)(2), 11(*-)(3), 12(+)(4),21(*)(5), 22(-+s)(6) ,111(*)(7), 112(8), 121(9) ,122(+-)(10) ,
211(*)(11), 212(12),221(*)(13),222(14),1111(-)(15),1112(16), 1121(*)(17), 1122(18), 1211(*)(19),
1212(+)(20),1221(-)(21),1222(22),2111(*)(23),2112(s)(24), 2121(25), 2122(26), 2211(27),
2212(-)(28), 2221(*)(29),2222(30),11111(*)(31),11112(32),11121(33),11122(34),11211(+)(35)...

For All *(primes Note. I am considering 1 as a prime here which is against convention)
1,11,21,111,211,221,1121,1211,2111,2221,11111...
Note Also most as if there purposely avoiding detection.
lets count how many 1s and 2s are in the numbers
(1,0),((2,0),(1,1)),((3,0),(2,1),(1,2)),((3,1),(3,1),(3,1),(1,3)),(5,0)
That's Interesting so far we have 1*1 digits long, 2*2 digits long, 3*3 digits long, 4*4 digits long.
I have no idea if this pattern continues but we have at least found something interesting here.

For All-(Triangle Numbers)
1,11,22,122,1111,1221,2212,11212...
lets count how many 1s and 2s are in the numbers.
(1,0),(2,0),(0,2),(1,2),(4,0),(1,3),(3,2)
These numbers kinda look like they are constantly trying to hold balance.

For All+(Triangle Numbers in 3D)
1,12,122,1212,11211...
This looks like counting further could yield a pattern certainly the digit length looks like it goes up one every number.

For all s(The Factorial Numbers)
1,2,22,2112...
It's hard to know with so little numbers if a pattern emerges.

The patterns seem hard to figure out. From analysis I have found that counting up to 2 and 3 in alternation seems to be useful to numbers and very interesting stuff might emerge. I'll go a bit deeper here.

So lets count tot the numbers up. In this case I will count both starting counting up to 2 and 3 so I'll produce 2 sets of numbers.

Set 2..
1(1),2(2),11(3),12(4),21(5),22(6),31(7),32(8),111(9),112(10),121(11),122(12),131(13),132(14),
211(15),212(16),221(17),222(18),231(19),232(20),1111(21),1112(22),1121(23),1122(24),1131(25),
1132(26),1211(27),1212(28),1221(29),1222(30),1231(31),1232(32),2111(33),2112(34),2121(35)...

Set 3..
1(1),2(2),3(3),11(4),12(5),13(6),21(7),22(8),23(9),111(10),112(11),113(12),121(13),122(14),
123(15),211(16),212(17),213(18),221(19),222(20),223(21),311(22),312(23),313(24),321(25),
322(26),323(27),1111(28),1112(29),1113(30),1121(31),1122(32),1123(33),1211(34),1212(35)...

So lets do the Analysis

For All(Primes)
For Set 2..
1,2,11,21,31,121,131,221,231,1121,1221,1231..
Lets do some digit counts of the 1's, 2's and 3's
((1,0,0),(0,1,0)),((2,0,0),(1,1,0),(1,0,1)),((2,1,0),(2,0,1),(1,2,0),(1,1,1)),((3,1,0),(2,2,0)(2,1,1)
I can't draw much here only that I notice the primes for each digit count equals the digit count +1 if this pattern continues I don't know but again this is of interest.

For Set 3..
1,2,3,12,21,112,121,212,221,312,1112,1121..
I notice when you compare numbers of each set with 21 at the end and no 3's you can match it with another prime on the other set. I don't know if this pattern continues

So lets catch up on what I have notice from all 3 sets that may or may not continue if billions of number were to be tested. I Personally am moving from Mathematica to Assembly in order to make some publications that people can test for themselves and to test patterns for myself.

So what I've noticed..
For Set 1 and Set 2
Prime counts from the first 2 sets appear to correlate with digit length.
For all sets.
There seems to be the Co-prime Information in either of the sets of a composite made of 2 co-primes but never in all of the sets.
For sets 2 and 3
When a 3 is in the number it is of interest.

I think this all demonstrates that counting without the zero does appear to give at least some information about complexity with in numbers and may well be worth further investigation.
I don't think one should rule out counting with the 0 as working with the zero and comparing results could yield interesting pattern also working with the 0 in some of the digit orders might allow people to find patterns that help with things like product function optimization in the future. However the full uses of doing things this way are yet unknown.