A367854 Indices at which record high values occur in A367821.

1, 2, 4, 6, 72, 75, 152, 518, 631, 1585, 2512, 4217, 5275, 13895, 14678, 53367, 177828, 464159, 1154782, 2154435, 3162278, 4641589, 8483429, 8576959, 13894955, 15848932, 21544347, 68129207, 74989421, 100000001, 114504757, 170125428, 517947468, 1000000001

Offset: 1

Author

William Hu, Dec 02 2023

Keywords

• nonn
• base

Comments

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

Examples

			From _Jon E. Schoenfield_, Dec 03 2023: (Start)
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:
.
     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)
		

Crossrefs

Cf. A367821.

Extensions

More terms from Jon E. Schoenfield, Dec 03 2023