Some numbers remain numbers when turned upside down (using digits 0125689, with no final 0's):
1, 2, 5, 6, 8, 9, 11, 12, … etc

IF a_{1}=1, a_{2}=2, a_{3}=5,... a_{8}=12,…, etc

a.Evaluate a_{100}

b. Evaluate a_{1000}

c. If a_{n}= 888, what is n?

The last digit can take any of 6 values. The remaining digits can take any of 7.

To get a(x),

1) express x as 6b + d, where d = x mod 6

2) then convert b to base 7, and append d

3) map the digits 0123456 to 0123689

For instance, x = 1,000,000 = 6*166,666 + 4

166,666 base 7 = 1262623

and 12626234 translates to 12929236,

so a(1,000,000) = 12,929,236

To go in the reverse direction, just reverse the process.

For example, a(x) = 888,888

This maps to 666666

Take all but the last digit (66666)

Convert to base 7 giving 16,806

Multiply by six and add the last digit: 6*16,806 + 6 = 100842,

so a(100,842) = 888,888