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'Kaprekar number'
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| Title of object: |
Kaprekar number |
| Canonical Name: |
KaprekarNumber |
| Type: |
Definition |
| Created on: |
2006-06-14 16:53:19 |
| Modified on: |
2006-06-23 15:15:41 |
| Classification: |
msc:11A63 |
Revision comment (for changes between this and next version):
| Changes for correction #8528 ('same again'). |
Preamble:
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Content:
An integer $n$ such that in a given base $b$ has $$n^2 = \sum_{i = 0}^{k - 1} d_ib^i$$ (where $d_x$ are digits, with $d_1$ the least significant digit and $d_k$ the most significant) such that $$\sum_{i = {k \over 2} + 1}^k d_ib^{i - {k \over 2} - 1} + \sum_{i = 1}^{k \over 2} d_ib^{i - 1} = n$$ if $k$ is even or $$\sum_{i = \lceil {k \over 2} \rceil}^k d_ib^{i - \lfloor {k \over 2} \rfloor - 1} + \sum_{i = 1}^{k \over 2} d_ib^{i - 1} = n$$ if $k$ is odd.
$b^x - 1$ for a natural $x$ is always a Kaprekar number in base $b$.
\begin{thebibliography}{1}
\bibitem{dk} D. R. Kaprekar, ``On Kaprekar numbers" {\it J. Rec. Math.} 13 (1980-1981), 81 - 82.
\end{thebibliography} |
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