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%I #27 Jan 19 2024 21:33:56
%S 1,2,4,6,72,75,152,518,631,1585,2512,4217,5275,13895,14678,53367,
%T 177828,464159,1154782,2154435,3162278,4641589,8483429,8576959,
%U 13894955,15848932,21544347,68129207,74989421,100000001,114504757,170125428,517947468,1000000001
%N Indices at which record high values occur in A367821.
%C Each term after a(1) = 1 is the smallest integer whose base-10 logarithm exceeds some ratio of integers N/D with D <= 21 = floor(1/(1 - log_10(9))); see Example section. - _Jon E. Schoenfield_, Dec 03 2023
%e From _Jon E. Schoenfield_, Dec 03 2023: (Start)
%e 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.
%e .
%e n a(n) log_10(a(n)) N/D log_10(a(n))*D
%e -- ------ -------------- ----- --------------
%e 2 2 0.301029995... 3/10 3.01029995...
%e 3 4 0.602059991... 3/5 3.01029995...
%e 4 6 0.778151250... 7/9 7.00336125...
%e 5 72 1.857332496... 13/7 13.00132747...
%e 6 75 1.875061263... 15/8 15.00049010...
%e 7 152 2.181843587... 24/11 24.00027946...
%e 8 518 2.714329759... 19/7 19.00030831...
%e 9 631 2.800029359... 14/5 14.00014679...
%e 10 1585 3.200029266... 16/5 16.00014633...
%e 11 2512 3.400019635... 17/5 17.00009817...
%e 12 4217 3.625003601... 29/8 29.00002880...
%e 13 5275 3.722222463... 67/18 67.00000435...
%e 14 13895 4.142858551... 29/7 29.00000985...
%e 15 14678 4.166666883... 25/6 25.00000130...
%e 16 53367 4.727272789... 52/11 52.00000068...
%e 17 177828 5.250000144... 21/4 21.00000057...
%e ...
%e 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:
%e .
%e k 177828^k
%e ------- ------------------
%e 1 1.7782800e+0000005
%e 2 3.1622799e+0000010
%e 3 5.6234188e+0000015
%e 4 1.0000013e+0000021
%e 5 1.7782824e+0000026
%e 6 3.1622840e+0000031
%e 7 5.6234263e+0000036
%e 8 1.0000027e+0000042
%e 9 1.7782847e+0000047
%e 10 3.1622882e+0000052
%e 11 5.6234338e+0000057
%e ...
%e 15 5.6234412e+0000078
%e 19 5.6234487e+0000099
%e 23 5.6234562e+0000120
%e ...
%e 1417539 8.9999657e+7442079
%e 1417543 8.9999776e+7442100
%e 1417547 8.9999896e+7442121
%e 1417551 9.0000015e+7442142
%e (End)
%Y Cf. A367821.
%K nonn,base
%O 1,2
%A _William Hu_, Dec 02 2023
%E More terms from _Jon E. Schoenfield_, Dec 03 2023