A364615 - OEIS

%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