# more from JHC re inconsummates

N. J. A. Sloane njas at research.att.com
Wed Jan 17 18:55:56 CET 2001

```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

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