In recreational mathematics, a Keith number or repfigit number (short for repetitive Fibonacci-like digit) is a number in the following integer sequence: 11, 15, 1E, 22, 2X, 31, 33, 44, 49, 55, 62, 66, 77, 88, 93, 99, XX, EE, 125, 215, 24X, 405, 42X, 654, 80X, 8X3, X59, 1022, 1662, 2044, 3066, 4088, 4X1X, 4XE1, 50XX, 8538, E18E, 17256, 18671, 24X78, 4718E, 517EX, 157617, 1X265X, 5X4074, 5XE140, 6E1449, 6E8515, ...

To determine whether an n-digit number N is a Keith number, create a Fibonacci-like sequence that starts with the n decimal digits of N, putting the most significant digit first. Then continue the sequence, where each subsequent term is the sum of the previous n terms. By definition, N is a Keith number if N appears in the sequence thus constructed.

As an example, consider the 3-digit number N = 125. The sequence goes like this:

1, 2, 5, 8, 13, 24, 43, 7X, 125, 226, ...

Because 125 appears in the sequence, 125 is seen to be indeed a Keith number.

The set of the Keith numbers has density zero. Whether or not there are infinitely many Keith numbers is currently a matter of speculation. Keith numbers are rare and hard to find. They can be found by exhaustive search, and no more efficient algorithm is known. There are about 0.E*(log(10)/log(2)) = 3.3526E7341879... Keith numbers between successive powers of 10.