[seqfan] 8/5 Sequence
zbi74583.boat at orange.zero.jp
zbi74583.boat at orange.zero.jp
Wed Sep 30 09:29:58 CEST 2009
Hi, Seqfan
8/5 Sequence :
IF x=Even THEN x=x/2
IF x=Odd AND
8*x=0 Mod 5 THEN x=(8*x+5)/5
8*x=1 Mod 5 THEN x=(8*x+4)/5
8*x=2 Mod 5 THEN x=(8*x+3)/5
8*x=3 Mod 5 THEN x=(8*x+2)/5
8*x=4 Mod 5 THEN x=(8*x+1)/5
S1 1,2,1,2,….
S2 3,5,9,15,25,41,66,33,53,85,….
S3 7,12,6,3,…
S4 11,18,9,….
Once Don Reble computed 2,000,000 th term = 852756... 564079; it has 46892
digits, where the first term is 107.
It seems to be divergent.
I think the most interesting thing is that 8/5 is rather small but it seems
to go infinity.
It is easy to generalize the sequence.
m/ n Sequence :
IF x=Even THEN x=x/2
IF x=Odd AND
m*x=0 Mod n THEN x=(m*x+n)/n
m*x=1 Mod n THEN x=(m*x+n-1)/n
m*x=2 Mod n THEN x=(m*x+n-2)/n
………....
m*x=n-1 Mod n THEN x=(m*x+1)/n
Example :
3/2 sequence is the same as 3x+1 sequence.
IF x=Even THEN x=x/2
IF x=Odd AND
3*x=0 Mod 2 THEN x=(3*x+2)/2 ….the case doesn’t exist
3*x=1 Mod 2 THEN x=(3*x+1)/2
Periodic :
7/5,9/5.11/5,12/5
4/3. 5/3
3/2
These m/n Sequence seem to be all periodic.
Only 8/5 Sequence is divergent in the area where m/n racio is around 3/2.
Obviously 10/5 Sequence is divergent.
Proof :
IF x=Even THEN x=x/2
IF x=Odd AND
10*x=0 Mod 5 THEN x=(10*x+5)/5=2*x+1 ….it is Odd.
So, S(n)=2*S(n-1)+1
And if m/n becomes much larger than 3 then it becomes divergent.
The number “3” is able to be a little smaller, it may be 2.7.
Yasutoshi
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