[seqfan] Re: Richard Guy's request for several new sequences
Alex M
timeroot.alex at gmail.com
Sun Jan 15 21:52:03 CET 2012
Correct me if I'm wrong, but I think these sequences would function fairly
differently in square bases, e.g. 4:
If the original number is m^2, the truncated number is n^2, and the removed
digit is k, then we have m^2 = 4*n^2 + k.
We can rewrite this, though, as (m-2n)*(m+2n) = k. Since m and n are bother
integers, either there's the obvious case k=0 and m=2n, which works for any
even square base 4, or if k=1, 2, or 3, then m-2n has to equal 1, and m+2n
is 1, 2, or 3, of which only m+2n=1 yields an integer solution - meaning
that besides the solution 1=1^2 -> 0=0^2, all such numbers are simply
dropping a zero off the end.
In higher bases, like 9 or 16, k has more possible values of course, so
there are more singular cases - but for each base these are only finite,
and then all other cases are yielded by dropping zeroes.
~6 out of 5 statisticians say that the
number of statistics that either make
no sense or use ridiculous timescales
at all has dropped over 164% in the
last 5.62474396842 years.
On Sun, Jan 15, 2012 at 12:20 PM, N. J. A. Sloane <njas at research.att.com>wrote:
>
> Dear SeqFans (and Richard)
>
> Richard Guy has been investigating recurrences
> satisfied by A023110 and related sequences - there
> are some surprising connections with continued
> factions. He has a very interesting preprint
> called "Neg and Reg", which he would probably
> be willing to send to anyone who was interested.
>
> A023110 is the central sequence: this lists squares
> which remain square when their last decimal digit is deleted.
> Their square roots are A204132. The truncated squares
> are A202303, and their square roots are (essentially) A031150.
>
> So for base 10 we have A023110, A204132, A202303, A031150.
> We need analogs of these 4 sequences for bases 2 through 9.
> Some will be in the OEIS already.
>
> Please help fill in the gaps.
>
>
> Best regards
> Neil
>
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