1, 2, 4, 8, 1, 3, 6, 1 ... is a non-cyclic series where a(n) represents the leading digit of 2^n.

a) How many ones are there within the first 3000 members?

b) Same question for a digit d, other than digit one.

Well, I guess my approximation to part (a) wasn't really an approximation. As Jer points out, the number of ones is necessarily the same as the number of digits in 2^3000, and this is exactly equal to 3000*log 2, rounded up to the next integer. Benford's law is still an approximation for the other values of d, although it is a pretty good one.

Benford's law:

predicted actual per xdog

1 30.1% 904

2 17.6% 528 529

3 12.5% 375 374

4 9.7% 291 291

5 7.9% 237 238

6 6.7% 201 201

7 5.8% 174 173

8 5.1% 153 155

9 4.6% 138 136