The dubs chain of an natural number n is a finite length sequence:
<s1, s2, s3, ..., sm>
where n has dubs si|s[sub]i/[sub] in base bi, with bi < bj when i < j. That is, the dubs chain is the finite list of dubs obtained by n, sorted in increasing order by the base. The ``base set'' of a natural number n is set of all bases in which n has dubs.
The ``dubs coefficient of n'', denoted ℡(n), is defined as the size of it's base set.
For example, 144 has dubs 2|2 in base71, 3|3 in base47, 4|4 in base35, 6|6 in base23, 8|8 in base17, 9|9 in base15, 0|0 in base12, 4|4 in base10, 0|0 in base6, 0|0 in base4, 0|0 in base3 and 0|0 in base2.
144 has dubs chain <0, 0, 0, 0, 4, 0, 9, 8, 6, 4, 3, 2>.
The base set of 144 is {2, 3, 4, 6, 10, 12, 15, 17, 23, 35, 47, 71},
and ℡(144) = 12.
A natural number n is a ``perfect dubs of order k'' if n has a dubs chain of <1, 2, 3, ..., k>.