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Eban Number


EbanNumbers

The eban numbers are the sequence of numbers whose names (in English) do not contain the letter "e" (i.e., "e" is "banned"). The name was coined by N. J. A. Sloane around 1990. 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.

The first few 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, ... (OEIS A006933); i.e., two, four, six, thirty, etc. These exclude one, three, five, seven, eight, nine, ten, eleven, twelve, etc.

In English, every odd number contains an "e," so all eban numbers are even (Hernandez et al. 2002-2003). In addition, eban numbers satisfy the following properties (Hernandez et al. 2002-2003).

1. There are gaps larger than any given number between eban numbers.

2. If a number of the form 2×10^k is an eban number, then 2×10^k+2 is also an eban number for any nonnegative integer k.

3. Let an "ebanie" be defined as 0 or one of the eban numbers 2, 4, 6, 30, 32, 34, 36, 40, 42, 44, 46, 50, 52, 54, 56, 60, 62, 64, or 66, and let an almost eban power be a power of 10 whose name contains exactly one "e." Then all eban numbers have the form sum_(k=0)^(n)a_k10^(3k), where a_k are ebanies that are not all zero and 10^3k are almost eban powers.

Binary representation of the eban numbers

A plot of the first few eban numbers represented as a sequence of binary bits is shown above. The top portion shows S_1 to S_(255), and the bottom shows the next 510 values.


See also

Aban Number, Iban Number, Number, Oban Number, Uban Number

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References

Hernandez, J. C.; Mex-Perera, C.; and Shepherd, S. J. "Characterization of Eban Numbers." J. Recr. Math. 31, 197-200, 2002-2003.Sloane, N. J. A. Sequence A006933/M1030 in "The On-Line Encyclopedia of Integer Sequences."

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Eban Number

Cite this as:

Weisstein, Eric W. "Eban Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EbanNumber.html

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