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The following table illustrates how the base-10 logarithm of each term from a(2) through a(17) is slightly larger than a ratio of integers N/D with D <= 21.
.
n a(n) log_10(a(n)) N/D log_10(a(n))*D
-- ------ -------------- ----- --------------
2 2 0.301029995... 3/10 3.01029995...
3 4 0.602059991... 3/5 3.01029995...
4 6 0.778151250... 7/9 7.00336125...
5 72 1.857332496... 13/7 13.00132747...
6 75 1.875061263... 15/8 15.00049010...
7 152 2.181843587... 24/11 24.00027946...
8 518 2.714329759... 19/7 19.00030831...
9 631 2.800029359... 14/5 14.00014679...
10 1585 3.200029266... 16/5 16.00014633...
11 2512 3.400019635... 17/5 17.00009817...
12 4217 3.625003601... 29/8 29.00002880...
13 5275 3.722222463... 67/18 67.00000435...
14 13895 4.142858551... 29/7 29.00000985...
15 14678 4.166666883... 25/6 25.00000130...
16 53367 4.727272789... 52/11 52.00000068...
17 177828 5.250000144... 21/4 21.00000057...
...
E.g., log_10(a(17)) = log_10(177828) slightly exceeds 21/4; 10^(21/4) = 10^5 * 10^(1/4) = 100000 * 1.77827941..., so 177828^k is slightly farther above the nearest lower power of 10 than 177828^(k-4) is. This near-periodic behavior of the mantissas, with their slow upward creep at every 4th exponent, explains why none of the mantissas of 177828^k begin with 9 until k gets very large:
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k 177828^k
------- ------------------
1 1.7782800e+0000005
2 3.1622799e+0000010
3 5.6234188e+0000015
4 1.0000013e+0000021
5 1.7782824e+0000026
6 3.1622840e+0000031
7 5.6234263e+0000036
8 1.0000027e+0000042
9 1.7782847e+0000047
10 3.1622882e+0000052
11 5.6234338e+0000057
...
15 5.6234412e+0000078
19 5.6234487e+0000099
23 5.6234562e+0000120
...
1417539 8.9999657e+7442079
1417543 8.9999776e+7442100
1417547 8.9999896e+7442121
1417551 9.0000015e+7442142
(End)
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