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%I #7 Aug 02 2023 13:46:22
%S 0,1,2,8,14,20,29,47,62,80,113,134,182,206,281,287,299,326,419,500,
%T 560,620,638,674,833,911,1271,1289,1376,1418,1583,1670,1814,2273,2753,
%U 3365,3794,4127,4160,4202,4280,4292,4538,4553,4646,4805,4952,4979,5105,5276
%N 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.
%C The average of the digits of 2^k is never exactly 9/2, because the sum of digits cannot be divisible by 3.
%C 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
%e k | 2^k | average of digits | distance from 9/2 | new minimum?
%e ---+-------+-------------------+-------------------+-------------
%e 0 | 1 | 1 | 7/2 | yes
%e 1 | 2 | 2 | 5/2 | yes
%e 2 | 4 | 4 | 1/2 | yes
%e 3 | 8 | 8 | 7/2 |
%e 4 | 16 | 7/2 | 1 |
%e 5 | 32 | 5/2 | 2 |
%e 6 | 64 | 5 | 1/2 |
%e 7 | 128 | 11/3 | 5/6 |
%e 8 | 256 | 13/3 | 1/6 | yes
%e 9 | 512 | 8/3 | 11/6 |
%e 10 | 1024 | 7/4 | 11/4 |
%e 11 | 2048 | 7/2 | 1 |
%e 12 | 4096 | 19/4 | 1/4 |
%e 13 | 8192 | 5 | 1/2 |
%e 14 | 16384 | 22/5 | 1/10 | yes
%Y Cf. A000079, A001370, A034887, A364606.
%K nonn,base
%O 1,3
%A _Pontus von Brömssen_ and _Jon E. Schoenfield_, Jul 29 2023