Perhaps the basis for further interesting sequences?

Chuck Seggelin seqfan at
Tue Jul 8 08:26:33 CEST 2008

1. Take a positive integer N.
2. Choose the largest digit of N, D.
3. Let B = D+1.
4. Treat N as a Base-B number and convert to base-10 yielding N'.
5. If N = N', you are done.

N will = N' if N is only one digit, or if N contains the digit 9, in which 
case you would be converting a base-10 number to base-10.

The record holders, (i.e. what value will produce a chain longer than the 
previous longest chain) already exists in the OEIS as A091049

But what about record-holding subchains where the same base is selected over 
and over?  For example:

[410245] (6)
--> [32501] (6)
--> [4501] (6)
--> [1045] (6)
--> [245] (6)
--> [101] (2)
--> [5]

410245, 32501, 4501, 1045, 245 is not a complete chain in that 245 can be 
"reduced" further, *but*, for this particular sequence each successive term 
can again be interpretted as a base-6 number.  Chains like this can 
sometimes be "pushed" forward, by picking the largest term and converting it 
to the desired base.  In this case it does not work, 410245 converted to 
base-6 yields 12443141 which by our rules would be treated as base 5 and 
yield the paltry sequence 12443141, 124796.

I haven't tried to record the sequence of single-base chain record holders 
for each base from 2 to 9, but here are the largest single-base chains I've 
found for each base thus far:

Longest base 2 chain was 2 terms starting at 1010
Longest base 3 chain was 3 terms starting at 21111
Longest base 4 chain was 3 terms starting at 133
Longest base 5 chain was 7 terms starting at 404044
Longest base 6 chain was 5 terms starting at 410245
Longest base 7 chain was 8 terms starting at 34406005
Longest base 8 chain was 9 terms starting at 170153
Longest base 9 chain was 12 terms starting at 33185141

Would there be any interest?  This would be 9 additional "base" sequences, 
and I understand that "base" sequences are not very popular.

-- Chuck Seggelin 

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