1 7-k = 4.3145..

2 105^(1/2)-k = 7.5615..

3 105^(1/3)-k = 2.0322..

4 2^(1/2)*7665^(1/4)-k = 10.5471..

**max**

5 2^(2/5)*7665^(1/5)-k = 5.2088..

6 2^(1/3)*7665^(1/6)-k = 2.9090..

7 2^(2/7)*7665^(1/7)-k = 1.6891..

8 2^(3/8)*7665^(1/8)-k = 1.2814..

9 2^(1/3)*7665^(1/9)-k = 0.7182..

10 2^(3/10)*3^(1/5)*2555^(1/10)-k = 0.6755..

11 2^(3/11)*3^(2/11)*2555^(1/11)-k = 0.3248..

12 2^(1/3)*21^(1/6)*365^(1/12)-k = 0.7362..

13 2^(5/13)*21^(2/13)*365^(1/13)-k = 0.5977..

14 2^(5/14)*21^(1/7)*365^(1/14)-k = 0.3304..

15 2^(1/3)*21^(2/15)*365^(1/15)-k = 0.1164..

16 2^(3/8)*21^(1/8)*365^(1/16)-k = 0.0580..

17 2^(7/17)*21^(2/17)*365^(1/17)-k = 0.0075..

18 2^(4/9)*21^(1/9)*365^(1/18)-k = -0.0366..

19 2^(9/19)*21^(2/19)*365^(1/19)-k = -0.0755..

20 2^(9/20)*21^(1/10)*365^(1/20)-k = -0.1977..

**max**

21 2^(11/21)*21^(1/7)*365^(1/21)-k = 0.2561..

**max**

22 2^(6/11)*21^(3/22)*365^(1/22)-k = 0.2050..

23 2^(12/23)*21^(3/23)*365^(1/23)-k = 0.0746..

24 2^(1/2)*21^(1/8)*365^(1/24)-k = -0.0396..

**max**

25 2^(12/25)*3^(4/25)*5^(2/25)*7^(3/25)*73^(1/25)-k = 0.1504..

Notice that 1-17 and 21-23 are

[Fred Lunnon was kind enough to point me to Chapter III of William Feller's

*positive*, while 18-20 and 24 are*negative*. Each one of these alternating-sign regimes has a maximum (distance from k): for instance, {20, 0.1977..} for the negative 18-20 regime. I have now calculated the start, end, and maximum for the first 27087 regimes (the final one incomplete because it ends beyond 3*10^9). Some regimes are quite lengthy, such as the positive 5418849-1434927964 and 1865143624->3*10^9. The so-far maximum in that final one is {2377934394, 0.00004194392..}, meaning that the geometric mean of 976 terms is closer to Khinchin than the geometric mean of 2377934394 terms![Fred Lunnon was kind enough to point me to Chapter III of William Feller's

*An Introduction to Probability Theory and its Applications*(Volume 1) as a means of understanding some of the mathematics involved in all this.]
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