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In mathematics, a Keith number or repfigit number 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
<math>N=\sum_{i=0}^{n-1} 10^i {d_i},<math>
a sequence <math>S_N<math> is formed with initial terms <math>d_{n-1}, d_{n-2},\ldots, d_1, d_0<math> and with a general term produced as the sum of the previous n terms. If the number N appears in the sequence <math>S_N<math>, then N is said to be a Keith number.
For example, taking 197 in such a way creates the sequence <math>1, 9, 7, 17, 33, 57, 107, 197, \ldots<math>. 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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