The "puzzle" with the geometric mean of consecutive gaps can be generalized: for n consecutive gaps the average ratio of the largest gap divided by the smallest gap appears to be as follows (rounded to three decimal places for n>=4):

Code:

n r_gm
1 1
2 4
3 8
4 12.641
5 17.757
6 23.249
7 29.052
8 35.121
9 41.423
10 47.928

I'm afraid these numbers will give me headaches.

1.8 n (0.38 + log n) is an asymptotic facsimile for n<=1000, but we want more than this.

What is it that I should ask myself?