more from JHC re inconsummates

[N. J. A. Sloane]

NJAS:  I'm posting this message (and the previous one)
in the hope that someone will send in the missing sequences.

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Date: Wed, 17 Jan 2001 12:08:12 -0500 (EST)
From: John Conway <conway at Math.Princeton.EDU>
To: "N. J. A. Sloane" <njas at research.att.com>
Subject: Re: Least inconsummate numbers.


    I think the most interesting of the sequences you haven't yet got
is what I call "the union sequence" (since it's the union of the
inconsummate numbers to all bases).  If you do bases up to 45, that
will get all terms below 45^2/2 > 1000, which should certainly be
enough to fill out three lines for you.

    Let me put the above ones into their own bases, to check my theorem:

   13 = 1101 in base 2
   17 = 122          3
   29 = 131          4
   16 = 31           5
   27 = 43           6
   30 = 42           7
   42 = 52           8
   46 = 51           9
   62 = 62          10 \ infinitely often it happens as here that two
   68 = 62          11 / successive answers look the same in their bases
   86 = 72          12
   92 = 71          13
  114 = 82          14 \ in fact this happens just for pairs of the form
  122 = 82          15 /  2,3 ; 10,11 ; 14,15 ; 18,19  (mod 20)
  147 = 93          16
  154 = 91          17

which is as far as I can go without two-digit "digits".  You'll see that
indeed the last digits for odd bases alternate between 1 and 2, while
those for even bases are 2 iff the base doesn't end in 6.

   Nice point that - for bases >4  congruent to

    0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 (mod 20)

the last digit of the least inconsummate number is

    2  1  2  2  2  1  x  2  2  1  2  2  2  1  2  2  x  1  2  2

where x >= 3.   This second digit is very small compared to the base,
the smallest bases for which it's

    1   2   3   4   5    6      8  ...    being

    2   3   6  36 106 1366 222106  ...

   (it can't be 7, or any larger number not of form [prime/2]).



    I do intend to write this stuff up, ...

     Regards, JHC