Review: A067581
David Wilson
davidwwilson at attbi.com
Sat Jun 1 01:58:15 CEST 2002
Regarding A067581:
In the name, "smallest integer" should be changed to "smallest positive
integer".
Regarding the question in the comment, I believe we can prove that A067581
contains
every positive integer except those containing all the digits 1 through 9
(which obviously
have no possible predecessor). A sketch of the proof:
Prove that for non-zero digit d, the sequence contains infinitely many
elements containing
the digit d. To do this, suppose no element beyond a(n) contains digit d.
Now let D be
the smallest rep-D integer not in (a(1),..,a(n)). After a(n), the sequence
can admit only
integers without digit d. It cannot admit an integer greater than D,
otherwise it would
have to admit D or a smaller integer in preference. Therefore, it must
admit only integers
smaller than D without digit d. It eventually exhausts these and is forced
to admit either
D or a smaller integer with digit d.
Now prove that for non-zero digit d, the sequence contains infinitely many
elements
without digit d. These would be the ones immediately following the infinite
number of
elements with digit d.
Now prove that for non-zero digit d, the sequence contains every rep-d
integer.
Let D be a rep-d integer. There are an infinitude of elements without digit
d, eventually
all suitable followers for these elements < D must be exhausted, and D must
be admitted.
Now prove that for every non-zero digit d, the sequence contains every
integer without
d. Every integer without d must at very worst eventually must be admitted
as the follower
to one of the infinite number of rep-d integers.
Thus the sequence omits every integer which contains all non-zero digits,
which have
no possible predecessor, and includes every other integer, which must be
missing some
non-zero digit.
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