[seqfan] Re: Factorial & Fibonacci number system analogues?

[Charles Greathouse]

I think the Zeckendorf expansion is as boring as the factorial base
representation here, since for any k there is a Fibonacci number which is a
multiple of k. Thus the Zeckendorf-flimsy numbers are just the
non-Fibonacci numbers A001690.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University


On Tue, Nov 19, 2013 at 5:46 AM, Antti Karttunen
<antti.karttunen at gmail.com>wrote:

> >
> > Message: 8
> > Date: Sun, 17 Nov 2013 21:54:56 -0500
> > From: Max Alekseyev <maxale at gmail.com>
> > To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> > Subject: [seqfan] Re: A125121, A005360.
> > Message-ID:
> >         <
> CAJkPp5NiQV6kfLtk+J+WxJ+3MAu+LG-03EwHXfMV+zTbvkUooA at mail.gmail.com>
> > Content-Type: text/plain; charset=UTF-8
> >
> > On Sun, Nov 17, 2013 at 8:32 PM, David Wilson <davidwwilson at comcast.net>
> wrote:
> >
> >> Likewise, the base-10 sturdy numbers would be those for which
> A007953(n) =
> >> A077196(n).
> >
> > These are A181862.
> >
> > Max
> >
>
> As far as I see, all numbers except factorials themselves are flimsy
> in factorial base representation ( http://oeis.org/A007623 ), as for
> any such number k, k! is one of its multiples and has the digit sum =
> 1 (= A034968(k!) ).
>
> However, what about Fibonacci number system (aka Zeckendorf Expansion)
> https://oeis.org/A014417 ?
> Is the concept well-defined there?
>
> Also, it would nice to investigate what factorial base analogues can
> be found for many of Eric Angelini's ideas, e.g. for the ones given
> below.
> (Overlapping multiplications, etc.)
>
>
> Cheers,
>
> Antti Karttunen
>
>
> > ------------------------------
>
> > Message: 5
> > Date: Mon, 18 Nov 2013 01:58:45 +0100
> > From: Eric Angelini <Eric.Angelini at kntv.be>
> > To: Sequence Discussion list <seqfan at list.seqfan.eu>,
> >         "eric.angelini at skynet.be" <eric.angelini at skynet.be>
> > Subject: [seqfan] Overlapping multiplications
> > Message-ID: <B8EDF14F-E5FF-4C9E-8CC2-9E1AAA1B7A4E at kntv.be>
> > Content-Type: text/plain; charset="utf-8"
> >
> >
> > Hello SeqFans,
> > Look at 59673468 and make a few
> > simple multiplications, digit by digit:
> >
> > 5.9=45
> >    9.6=54
> >      6.7=42
> >        7.3=21
> >         3.4=12
> >           4.6=24
> >             6.8=48
> >
> > From the second line on, each result
> > starts with the last digit of the previous
> > result. We could thus shrink the above
> > staircase like this:
> >
> > 59673468
> > 45421248
> >
> > ... where we see that the first 2-digit
> > multiplication 5x9 produces 45 (on
> > the next line), the second 2-digit
> > multiplication 9x6 produces 54
> > (overlapping the previous 45), the
> > third multiplication 6x7 produces 42
> > (overlapping the previous 54), etc.
> >
> > What would a seq of such "overlapping multiplication" (OM)
> > integers be?
> >
> > Two more remarks:
> >
> > a) 581 is such an OM integer if we allow the notation 8x1=08, as the
> > results 5x8=40 and 8x1=08 will
> > correctly overlap:
> >
> > 581
> > 408
> >
> > b) If we allow in the same way that
> > 0x0=00 then 5081 will also be
> > accepted as an OM integer, as the
> > three results 00,00 and 08 will
> > correctly overlap too:
> >
> > 5081
> > 0008
> >
> > Let?s start the OM seq with a(1)=100:
> >
> > OM = 100,101,102,103,104,105,106,107,108,
> > 109,200,201,202,203,204,205,206,207,
> > 208,209,251,264,276,277,288,299,300,
> > 301,302,303,304,305,306,307,308,309,
> > 345,346,347,372,385,386,398,400,401,
> > 402,403,404,405,406,407,408,409,437,
> > 438,439,451,467,468,483,497,500,501,
> > 502,503,504,505,506,507,508,509,521,
> > 522,523,524,541,542,561,578,581,596,...
> >
> > The OM integers are easy to find and
> > the OM seq is infinite -- as there is at
> > least one looping OM:
> >
> > 3467(3)
> > 1242(1)
> >
> > The 00 trick is another proof of OM being
> > infinite:
> >
> > 10000000000... is an OM
> > 00000000000...
> >
> > 102030405060708090 is an OM
> > 000000000000000000
> >
> > One could decide that the OM seq
> > should begin with 10 -- no overlapping here, but no contradiction
> > either (OM would then start with all
> > integers between 10 and 99, including 10 and 99).
> >
> > Best,
> > ?.
> > --------------------------------------
> > P.-S.
> > I'm working now on the OA seq (overlapping addition):
> >
> > OA = 1,2,3,-->97,98,99,190,280,281,
> > 291,292,293,294,295,296,297,298,299,
> > 370,371,372,382,383,384,385,386,387,
> > 388,389,460,461,462,463,474,475,476,
> > 477,478,479,550,551,552,553,554,640,
> > 641,642,643,644,645,656,657,658,659,
> > 730,731,732,733,734,735,736,747,748,
> > 749,820,821,822,823,824,825,826,827,
> > 910,911,912,913,914,915,916,917,918,
> > 929,2910,...
> > (is 2910 the OA term after 929?)
> > Best,
> > ?.
> >
> >
> > ------------------------------
>
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