In recreational mathematics, a Keith number or repfigit number (short for repetitive Fibonacci-like digit) is a natural number
digits such that when a sequence is created such that the first
and each subsequent term is the sum of the previous
is part of the sequence.
Keith numbers were introduced by Mike Keith in 1987.
[1] They are computationally very challenging to find, with only about 100 known.
We define the sequence
by a linear recurrence relation.
For example, 88 is a Keith number in base 6, as and the entire sequence and
Whether or not there are infinitely many Keith numbers in a particular base
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.
[2] According to Keith, in base 10, on average
Keith numbers are expected between successive powers of 10.
14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909, 31331, 34285, 34348, 55604, 62662, 86935, 93993, 120284, 129106, 147640, 156146, 174680, 183186, 298320, 355419, 694280, 925993, 1084051, 7913837, 11436171, 33445755, 44121607, 129572008, 251133297, ...[4] In base 2, there exists a method to construct all Keith numbers.
[3] The Keith numbers in base 12, written in base 12, are where ᘔ represents 10 and Ɛ represents 11.
A Keith cluster is a related set of Keith numbers such that one is a multiple of another.
These are possibly the only three examples of a Keith cluster in base 10.
[5] The example below implements the sequence defined above in Python to determine if a number in a particular base is a Keith number: