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A364615
Numbers k such that the average of the decimal digits of 2^k is closer to 9/2 (the expected average for random digits) than for any smaller power of 2.
0
0, 1, 2, 8, 14, 20, 29, 47, 62, 80, 113, 134, 182, 206, 281, 287, 299, 326, 419, 500, 560, 620, 638, 674, 833, 911, 1271, 1289, 1376, 1418, 1583, 1670, 1814, 2273, 2753, 3365, 3794, 4127, 4160, 4202, 4280, 4292, 4538, 4553, 4646, 4805, 4952, 4979, 5105, 5276
OFFSET
1,3
COMMENTS
The average of the digits of 2^k is never exactly 9/2, because the sum of digits cannot be divisible by 3.
Conjecture: for each term k > 1, digitsum(2^k) - (9/2)*number_of_digits(2^k) = 1/2 if k is odd, -1/2 if k is even. - Jon E. Schoenfield, Jul 30 2023
EXAMPLE
k | 2^k | average of digits | distance from 9/2 | new minimum?
---+-------+-------------------+-------------------+-------------
0 | 1 | 1 | 7/2 | yes
1 | 2 | 2 | 5/2 | yes
2 | 4 | 4 | 1/2 | yes
3 | 8 | 8 | 7/2 |
4 | 16 | 7/2 | 1 |
5 | 32 | 5/2 | 2 |
6 | 64 | 5 | 1/2 |
7 | 128 | 11/3 | 5/6 |
8 | 256 | 13/3 | 1/6 | yes
9 | 512 | 8/3 | 11/6 |
10 | 1024 | 7/4 | 11/4 |
11 | 2048 | 7/2 | 1 |
12 | 4096 | 19/4 | 1/4 |
13 | 8192 | 5 | 1/2 |
14 | 16384 | 22/5 | 1/10 | yes
CROSSREFS
KEYWORD
nonn,base
AUTHOR
STATUS
approved