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, 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

Links

Table of n, a(n) for n=1..50.

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

Cf. A000079, A001370, A034887, A364606.

Sequence in context: A101959 A241003 A133229 * A161330 A046940 A046939

Adjacent sequences: A364612 A364613 A364614 * A364616 A364617 A364618

Keyword

nonn,base

Author

Pontus von Brömssen and Jon E. Schoenfield, Jul 29 2023

Status

approved