[seqfan] Overlapping multiplications
Eric Angelini
Eric.Angelini at kntv.be
Mon Nov 18 01:58:45 CET 2013
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,
É.
--------------------------------------
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,
É.
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