Hmmm ... yes, a bit of similarity in runs of initial decimal digits:

...

607

1279

2203

2281

3217

4253

4423

9689

...

6972593

13466917

20996011

24036583

25964951

30402457

32582657

37156667

43112609

(?)

But why should that mean anything other than a coincidence within a larger demonstration of Benford's Law (

http://mathworld.wolfram.com/BenfordsLaw.html or

http://en.wikipedia.org/wiki/Benford's_law) over a set of numbers (_all_ known Mersenne-prime exponents, not just two specially-picked subsets) whose distribution is expected to be related to logarithms?

Looking at all the known Mersenne-prime exponents with 1-7 decimal digits (we don't yet have a complete census for 8-decimal-digit exponents), the counts by initial digit are:

1: 12

2: 7

3: 4

4: 3

5: 2

6: 3

7: 2

8: 3

9: 2

Of the 38 exponents in that range,

12/38 = 32% start with "1",

7/38 = 18% start with "2",

4/38 = 11% start with "3".

Compare that to the ideal Benford distribution of

30.1% "1"s,

17.6% "2"s, and

12.5% "3"s.

Pretty close for such a small sample (N = 38), eh? And -- Benford's Law comes with a logical mathematical explanation -- no guessing needed!