keith number

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

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.

External link

Keith numbers (sequence A007629) in OEIS

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