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sci.math.research > have you seen this sequence?
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David Madore   Oct 24 2004, 8:00 pm     show options
Newsgroups: sci.math.research
From: david.mad...@ens.fr (David Madore) - Find messages by this author
Date: Mon, 25 Oct 2004 00:00:09 +0000 (UTC)
Local: Sun,Oct 24 2004 8:00 pm
Subject: have you seen this sequence?
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Essentially, I'm wondering whether the following sequence, which I
encountered by chance, seems sufficiently interesting to be added to
Sloane's encyclopaedia, and incidentally whether someone has met it
before or has anything worth while to say about it.

It is probably easier to show how it is constructed rather than to try
to describe it in words (here for the first 30 terms):

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
1 2 1 2 3 3 1 2 1 2 3 3 1 2 1 2 3 3 1 2 1 2 3 3 1 2 1 2 3 3
1 2 1 2 3 3 1 2 4 4 3 4 1 2 1 2 3 3 1 2 4 4 3 4 1 2 1 2 3 3
1 2 1 2 3 3 1 2 4 4 3 4 1 2 5 5 3 5 1 2 4 5 3 4 1 2 1 2 3 3
1 2 1 2 3 3 1 2 4 4 3 4 1 2 5 5 3 5 1 2 4 5 3 4 6 6 1 2 6 3

start with a sequence of 1's on the first line; then replace every
other 1 by a 2 to form the second line; then replace every third 1 by
a 3 and every third 2 by a 3 to form the third line; then replace
every fourth 1, every fourth 2 and every fourth 3 by a 4; and so on.
The sequence I speak of is the limit of these: it is obvious that,
whatever the (finite) number of terms that are to be produced, after a
certain number of "lines" the terms in question aren't affected any
more.

<URL: http://www.eleves.ens.fr:8080/home/madore/.misc/seq.png > gives
an idea of what the sequence looks like for a largish number of terms
(2520, chosen because it is the LCM of the integers from 1 through
10).  Experimentally it seems that the supremum of the n first terms
of the sequence grows like sqrt(4n/3).  One could also ask for the
density of 1's among the n first term, which can probably be computed
since the 1's are selected from a kind of sieving process (remove
every other 1, then every third, then every fourth, and so on),
remeniscent to that for primes or "lucky numbers".

So, is this a worthy candidate for inclusion in Sloane?

--
     David A. Madore
    (david.mad...@ens.fr,
     http://www.dma.ens.fr/~madore/ )

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