# Keith number

Given a base $b$ integer

 $n=\sum_{i=1}^{k}d_{i}b^{i-1}$

where $d_{1}$ is the least significant digit and $d_{k}$ is the most significant, construct the sequence $a_{1}=d_{k},\ldots a_{k}=d_{1}$, and for $m>k$,

 $a_{m}=\sum_{i=1}^{k}a_{m-i}.$

If there is an $x$ such that $a_{x}=n$, then $n$ is a Keith number or repfigit number.

In base 10, the first few Keith numbers below ten thousand are: 14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909 (see A007629 in Sloane’s OEIS for a longer listing). 47 is a base 10 Keith number because it is contained the Fibonacci-like recurrence started from its base 10 digits: 4, 7, 11, 18, 29, 47, etc.

## References

• 1 M. Keith, “Repfigit Numbers” J. Rec. Math. 19 (1987), 41 - 42.
Title Keith number KeithNumber 2013-03-22 16:00:20 2013-03-22 16:00:20 PrimeFan (13766) PrimeFan (13766) 5 PrimeFan (13766) Definition msc 11A63 repfigit number