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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