[seqfan] Re: plastic number base

[Graeme McRae]

Yes, there is a relation between A205579 and the numbers that have a
"sparse" representation in "base plastic", where the plastic number is
p=1.324717957...  A205579 lists 10717, for example, because 10717 is
ever-so-slightly larger than p^33, so the next digit in the "base plastic"
representation of 10717 is a tiny number -- p^-18.

The "base plastic" representation of an integer is always sparse, because
the "standard form" of such a representation always has at least four zeros
between every pair of ones.  That's because
010001(base p) = 100000 (base p),
0100100000 (base p) = 1000000001 (base p),
0101000 (base p) = 1000001 (base p),
0011 (base p) = 1000 (base p),
00200000 (base p) = 10000001 (base p)
The right-hand-side of each of these identities has the 1's more "spread
out" than the left side, so in each case where the 1's are separated by
fewer than 4 zeros, one of these identities can be used to spread out the
ones, until the number is in "standard form".

--Graeme McRae
Palmdale, CA


On Tue, Jul 9, 2013 at 12:31 PM, L. Edson Jeffery <lejeffery2 at gmail.com>wrote:

> >Is there any signicance to these numbers?
>
> Cf. A205579 and A109377 which contain some of your terms.
>
> Ed Jeffery
>
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