Keith number

In mathematics, a Keith number or repfigit number (short for repetitive Fibonacci-like digit) is an integer that appears as a term in a linear recurrence relation with initial terms based off its own digits. Given an n-digit number

$N=\sum_{i=0}^{n-1} 10^i {d_i},$

a sequence $S N$ is formed with initial terms $d_{n-1}, d_{n-2},\ldots, d_1, d_0$ and with a general term produced as the sum of the previous n terms. If the number N appears in the sequence $S N$ , then N is said to be a Keith number.

For example, taking 197 in such a way creates the sequence $1, 9, 7, 17, 33, 57, 107, 197, \ldots$ . The first few Keith numbers are

14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909

Whether or not there are infinitely many Keith numbers is currently a matter of speculation. There are only 71 Keith numbers below 10¹⁹, making them much rarer than prime numbers.

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Keith numbers (sequence A007629) in OEIS

Categories: Integer sequences

Last updated: 10-15-2005 13:12:16

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