[seqfan] Factorial & Fibonacci number system analogues?

[Antti Karttunen]

>
> 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,
> ?.
>
>
> ------------------------------