[seqfan] Re: Bi-digital multiplications (concatenated)

Frank Adams-Watters franktaw at netscape.net
Mon Nov 24 01:30:07 CET 2014


1, 2, ..., 9 are not fixed points. They all immediately lead to 0, 
which is a fixed point.

(With only one digit, there are no pairs, so the concatenation of the 
products is empty. The empty string is another representation for zero.)

Franklin T. Adams-Watters

P.S. It's multiply, not multiplicate.

-----Original Message-----
From: Eric Angelini <Eric.Angelini at kntv.be>
To: Eric Angelini <eric.angelini at skynet.be>; Sequence Discussion list 
<seqfan at list.seqfan.eu>
Sent: Sun, Nov 23, 2014 5:12 pm
Subject: [seqfan] Bi-digital multiplications (concatenated)



Hello SeqFans,
Here is how we "bi-digital" multiplicate;
1) we always read a number N (like 1235) from left to right
2) we successively multiplicate the pairs of digits we encounter in N
3) we concatenate the results:
1235-->2615
(2615 is the concatenation of 1.2, 2.3 and 3.5 that is 2, 6 and 15).

We now iterate to see what happens:

2615--> 1265--> 21230--> 2260--> 4120--> etc.

I guess that the iterating process has three outcomes:
1) fixed point 0, or 1, or 2, or... or 8, or 9
2) infinite expansion (with or without visible patterns)
3) loop

I've found (1) and (2) but not (3)...

Examples:
184-->832-->246-->824-->168-->648-->2432-->8126-->8212-->1622-->6124-->62
8-->1216-->226-->412-->42-->8
END

185-->840-->320-->60-->0 END

186-->848-->3232-->666-->3636-->181818-->88888-->64646464-->2424242424242
4-->8888888888888-->
INF

If this is of interest, one could submit at least a dozen or so 
sequences to the
OEIS:

a) integers ending on 0
A=0,10,20,25,30,40,45,50,52,54,55,56,58,59,60,...
[this is not http://oeis.org/A034048 as 
239-->627-->1214-->224-->48-->32-->6
here but -->0 in the OEIS seq "multiplicative digital root value 0"]
b) integers ending on 1
B=1,11,111,1111,11111,... [already in the OEIS]
c) integers ending on 2
C=2,12,21,26,34,37,43,62,
... [this is not http://oeis.org/A034049, "multiplicative digital root 
value 2"]
j) integers ending on 9
J=9,19,33,91,119,133,191,... [this is not http://oeis.org/A034056]
k) integers expanding for ever
K=186, ?, ?, ?,...
[266 is a member of K, though no immediately visible pattern arises]
l) integers ending in a loop
[are there any?]
m) integers with no predecessor
[like 281]
n) integers with exactly one predecessor [23, for instance: 23<--213]
o) integers with exactly two predecessors [189<--291 or 189<--633]
etc.

P.-S.
We consider that 257,for instance, ends on the fixed point 0 [although 
all
intermediary integers "do not exist" (because of leading zeroes 
somewhere in the
iteration process: 257-->1035-->0015-->005-->00-->0)]
Best,
É.




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