An iban number is a positive integer whose conventional English cardinal name does not contain the letter "i" (i.e., "i" is banned), by analogy with the eban numbers. The first few are 1, 2, 3, 4, 7, 10, 11, 12, 14, 17, 20, 21, 22, 23, 24, 27, 40, ... (OEIS A089589, with its initial zero omitted). Note that this definition is imprecise insofar as special names are sometimes assigned to a few large numbers that do not follow the usual rules for the naming of such numbers.
Ignoring "special" number names such as googol and googolplex, the conventional short-scale name
of every power 
with integer exponent 
contains the suffix "-illion," so there are a finite number of iban numbers.
In fact, there are a total of 30275 of them, the largest of which is 777777.
A plot of the first few iban numbers represented as a sequence of binary bits is shown above. The top portion shows the first 255 values, and the bottom shows the next 510 values.
