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# Keith number

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

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.

Synonym:

repfigit number

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

11A63*no label found*

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## Comments

## Repsylvits

Just wondring if anyone here has studied repsylvits (kind of like repfigits, but using the Silvester sequence).

In base 10 their might only be three: 13 91 adn 2551.

1, 3, 4, 13

9, 1, 10, 91

2, 5, 5, 1, 51, 2551

Ive looked upto 10^5.