First of all, I believe it makes it much easier to make this table before starting:

1 - 0

2 - 1

3 - 10

4 - 2

5 - 100

6 - 11

7 - 1000

8 - 3

9 - 20

10 - 101

11 - 10000

12 - 12

13 - 100000

14 - 1001

15 - 110

16 - 4

17 - 1000000

18 - 21

19 - 10000000

20 - 102

21 - 1010

22 - 10001

23 - 100000000

24 - 13

25 - 200

26 - 100001

27 - 30

28 - 1002

Now. **Ino**’s post gives us all primes (1, 10, 100, 1000, 10000, etc.) all powers of primes (1000, 2000, 3000, etc.) and all numbers expressible as (2^n)*p where n is an integer and p is a prime (10000, 10001, 10002, 10003, etc.)

I think the next step is to figure out the pattern for all multiples of a given prime.

The known multiples of primes are:

Twos:

2 - 1

4 - 2

6 - 11

8 - 3

10 - 101

12 - 12

14 - 1001

16 - 4

18 - 21

20 - 102

22 - 10001

24 - 13

26 - 100001

28 - 1002

Threes:

3 - 10

6 - 11

9 - 20

12 - 12

15 - 110

18 - 21

21 - 1010

24 - 13

27 - 30

Fives:

5 - 100

10 - 101

15 - 110

20 - 102

25 - 200

Sevens:

7 - 1000

14 - 1001

21 - 1010

28 - 1002

Elevens:

11 - 10000

22 - 10001

Thirteens:

13 - 100000

26 - 100001

Note that a number composed of two primes must fit two patterns (i.e. 15 fits the 3-pattern and the 5-pattern). After the prime itself, replace the final zero with a 1 to get the next number (2p), with the exception of 4=2*2, expressed as “2” because it’s a power. Beyond that, I feel like I can kind of “see” the pattern, but I don’t know exactly what it is. Hopefully all my tables will help others. Meanwhile, I’m still working on this.