[seqfan] Re: Sharing no digit business ((ADD)END(UM))
franktaw at netscape.net
franktaw at netscape.net
Sun Apr 5 05:02:39 CEST 2009
This is no coincidence; it is easily established by induction. Let n
be the smallest integer such that a(a(n)) != n; call a(n) m.
Manifestly n and m have no digit in common, so there are two cases:
1) n is the value of some a(k) with k < m. But then k would have been
a preferred choice for a(n).
2) a(m) = k, with k < n. But then by induction, we must have a(a(k)) =
k, and so a(k) = m. But this contradicts a(n) = m.
Note that the only property of the "shares no digit" relation used in
this proof is that it is symmetric, so this result also applies in a
number of similar cases.
Franklin T. Adams-Watters
-----Original Message-----
From: David Wilson <davidwwilson at comcast.net>
....
1. a(n) = smallest unseen positive integer such that a(n) shares no
digit with n.
....
I also found something rather amusing. As far as I calculated (1), I
found that a(a(n)) = n.
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