# Ban number

In recreational mathematics a ban number is a number whose English name does not contain a specified letter. For example, an eban number is a number that does not contain the letter 'e' in its English name.

## Aban numbers

Aban numbers are defined as numbers that do not contain the letter 'a' when spelled in English. The letter 'a' in the word 'and' is not counted. This definition is not definitive as some large numbers do not follow the guidelines for naming of numbers.

From this definition, one understands that numbers including the word "thousand" are not aban numbers, but all smaller numbers are. The first aban numbers are: 1-999, 1000000-1000999, 2000000-2000999, ... etc.

## Eban numbers

Eban numbers are defined as numbers that do not contain the letter 'e' when spelled in English. This definition is not definitive as some large numbers don't follow the guidelines for naming of numbers. The name of this term was created by Neil James Alexander Sloane in 1990.

Because every odd number contains an e, all eban numbers are even. The first eban numbers are:

2, 4, 6, 30, 32, 34, 36, 40, 42, 44, 46, 50, 52, 54, 56, 60, 62, 64, 66, 2000, 2002, 2004, ... Template:OEIS

Some properties of eban numbers:

1. A gap larger than any given number can be found between two consecutive eban numbers.
2. If $2\cdot 10^k$ is an eban number then $2\cdot 10^k+2$ is also an eban number for $k\in N$.

## Iban numbers

Iban numbers are defined as numbers that do not contain the letter 'i' when spelled in English. This definition is not definitive as some large numbers don't follow the guidelines for naming of numbers e.g. googol and googolplex.

The first iban numbers are:

1, 2, 3, 4, 7, 10, 11, 12, 14, 17, 20, 21, 22, 23, 24, 27, 40, ... Template:OEIS

As every power of 10 with an exponent greater of 5 contains the suffix -illion the list of iban numbers is finite. There are exactly 30275 iban numbers and the largest one is 777777.

## Oban numbers

Oban numbers are defined as numbers that do not contain the letter 'o' when spelled in English. This definition is not definitive as some large numbers don't follow the guidelines for naming of numbers.

The first oban numbers are:

3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 23, 25, 26, ... Template:OEIS

Because every number name for a power of 10 greater than 3 contains the suffix thousand or -illion, the list of oban numbers is finite. There are exactly 454 oban numbers and the largest one is 999.

## Uban numbers

Uban numbers are defined as numbers that do not contain the letter 'u' when spelled in English. This definition is not definitive as some large numbers don't follow the guidelines for naming of numbers.

The first uban numbers are:

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, ... Template:OEIS

## References

Hernandez, J. C.; Mex-Perera, C.; and Shepherd, S. J. "Characterization of Eban Numbers." J. Recr. Math. 31, 197-200, 2002-2003.