Keith Number/Examples/251,133,297
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Examples of Keith Number
$251 \, 133 \, 297$ is a Keith number:
- $2, 5, 1, 1, 3, 3, 2, 9, 7, 33, 64, 123, 245, 489, 975, 1947, 3892, 7775,$
- $15 \, 543, 31 \, 053, 62 \, 042, 123 \, 961, 247 \, 677, 494 \, 865, 988 \, 755,$
- $1 \, 975 \, 563, 3 \, 947 \, 234, 7 \, 886 \, 693, 15 \, 757 \, 843, 31 \, 484 \, 633,$
- $62 \, 907 \, 224, 125 \, 690 \, 487, 251 \, 133 \, 297, \ldots$
Proof
By definition of Keith number, we create a Fibonacci-like sequence $K$ from $\left({2, 5, 1, 1, 3, 3, 2, 9, 7}\right)$:
\(\ds K_0\) | \(=\) | \(\ds 2\) | ||||||||||||
\(\ds K_1\) | \(=\) | \(\ds 5\) | ||||||||||||
\(\ds K_2\) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds K_3\) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds K_4\) | \(=\) | \(\ds 3\) | ||||||||||||
\(\ds K_5\) | \(=\) | \(\ds 3\) | ||||||||||||
\(\ds K_6\) | \(=\) | \(\ds 2\) | ||||||||||||
\(\ds K_7\) | \(=\) | \(\ds 9\) | ||||||||||||
\(\ds K_8\) | \(=\) | \(\ds 7\) | ||||||||||||
\(\ds K_9\) | \(=\) | \(\ds K_0 + K_1 + K_2 + K_3 + K_4 + K_5 + K_6 + K_7 + K_8\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 + 5 + 1 + 1 + 3 + 3 + 2 + 9 + 7\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 33\) | ||||||||||||
\(\ds K_{10}\) | \(=\) | \(\ds K_1 + K_2 + K_3 + K_4 + K_5 + K_6 + K_7 + K_8 + K_9\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 5 + 1 + 1 + 3 + 3 + 2 + 9 + 7 + 33\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 64\) | ||||||||||||
\(\ds K_{11}\) | \(=\) | \(\ds K_2 + K_3 + K_4 + K_5 + K_6 + K_7 + K_8 + K_9 + K_{10}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 + 1 + 3 + 3 + 2 + 9 + 7 + 33 + 64\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 123\) | ||||||||||||
\(\ds K_{12}\) | \(=\) | \(\ds K_3 + K_4 + K_5 + K_6 + K_7 + K_8 + K_9 + K_{10} + K_{11}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 + 3 + 3 + 2 + 9 + 7 + 33 + 64 + 123\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 245\) | ||||||||||||
\(\ds K_{13}\) | \(=\) | \(\ds K_4 + K_5 + K_6 + K_7 + K_8 + K_9 + K_{10} + K_{11} + K_{12}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 + 3 + 2 + 9 + 7 + 33 + 64 + 123 + 245\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 489\) | ||||||||||||
\(\ds K_{14}\) | \(=\) | \(\ds K_5 + K_6 + K_7 + K_8 + K_9 + K_{10} + K_{11} + K_{12} + K_{13}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 + 2 + 9 + 7 + 33 + 64 + 123 + 245 + 489\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 975\) | ||||||||||||
\(\ds K_{15}\) | \(=\) | \(\ds K_6 + K_7 + K_8 + K_9 + K_{10} + K_{11} + K_{12} + K_{13} + K_{14}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 + 9 + 7 + 33 + 64 + 123 + 245 + 489 + 975\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1947\) | ||||||||||||
\(\ds K_{16}\) | \(=\) | \(\ds K_7 + K_8 + K_9 + K_{10} + K_{11} + K_{12} + K_{13} + K_{14} + K_{15}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 9 + 7 + 33 + 64 + 123 + 245 + 489 + 975 + 1947\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3892\) | ||||||||||||
\(\ds K_{17}\) | \(=\) | \(\ds K_8 + K_9 + K_{10} + K_{11} + K_{12} + K_{13} + K_{14} + K_{15} + K_{16}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7 + 33 + 64 + 123 + 245 + 489 + 975 + 1947 + 3892\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7775\) | ||||||||||||
\(\ds K_{18}\) | \(=\) | \(\ds K_9 + K_{10} + K_{11} + K_{12} + K_{13} + K_{14} + K_{15} + K_{16} + K_{17}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 33 + 64 + 123 + 245 + 489 + 975 + 1947 + 3892 + 7775\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 15 \, 543\) | ||||||||||||
\(\ds K_{19}\) | \(=\) | \(\ds K_{10} + K_{11} + K_{12} + K_{13} + K_{14} + K_{15} + K_{16} + K_{17} + K_{18}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 64 + 123 + 245 + 489 + 975 + 1947 + 3892 + 7775 + 15 \, 543\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 31 \, 053\) | ||||||||||||
\(\ds K_{20}\) | \(=\) | \(\ds K_{11} + K_{12} + K_{13} + K_{14} + K_{15} + K_{16} + K_{17} + K_{18} + K_{19}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 123 + 245 + 489 + 975 + 1947 + 3892 + 7775 + 15 \, 543 + 31 \, 053\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 62 \, 042\) | ||||||||||||
\(\ds K_{21}\) | \(=\) | \(\ds K_{12} + K_{13} + K_{14} + K_{15} + K_{16} + K_{17} + K_{18} + K_{19} + K_{20}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 245 + 489 + 975 + 1947 + 3892 + 7775 + 15 \, 543 + 31 \, 053 + 62 \, 042\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 123 \, 961\) | ||||||||||||
\(\ds K_{22}\) | \(=\) | \(\ds K_{13} + K_{14} + K_{15} + K_{16} + K_{17} + K_{18} + K_{19} + K_{20} + K_{21}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 489 + 975 + 1947 + 3892 + 7775 + 15 \, 543 + 31 \, 053 + 62 \, 042 + 123 \, 961\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 247 \, 677\) | ||||||||||||
\(\ds K_{23}\) | \(=\) | \(\ds K_{14} + K_{15} + K_{16} + K_{17} + K_{18} + K_{19} + K_{20} + K_{21} + K_{22}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 975 + 1947 + 3892 + 7775 + 15 \, 543 + 31 \, 053 + 62 \, 042 + 123 \, 961 + 247 \, 677\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 494 \, 865\) | ||||||||||||
\(\ds K_{24}\) | \(=\) | \(\ds K_{15} + K_{16} + K_{17} + K_{18} + K_{19} + K_{20} + K_{21} + K_{22} + K_{23}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1947 + 3892 + 7775 + 15 \, 543 + 31 \, 053 + 62 \, 042 + 123 \, 961 + 247 \, 677 + 494 \, 865\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 988 \, 755\) | ||||||||||||
\(\ds K_{25}\) | \(=\) | \(\ds K_{16} + K_{17} + K_{18} + K_{19} + K_{20} + K_{21} + K_{22} + K_{23} + K_{24}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3892 + 7775 + 15 \, 543 + 31 \, 053 + 62 \, 042 + 123 \, 961 + 247 \, 677 + 494 \, 865 + 988 \, 755\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 \, 975 \, 563\) | ||||||||||||
\(\ds K_{26}\) | \(=\) | \(\ds K_{17} + K_{18} + K_{19} + K_{20} + K_{21} + K_{22} + K_{23} + K_{24} + K_{25}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7775 + 15 \, 543 + 31 \, 053 + 62 \, 042 + 123 \, 961 + 247 \, 677 + 494 \, 865 + 988 \, 755 + 1 \, 975 \, 563\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \, 947 \, 234\) | ||||||||||||
\(\ds K_{27}\) | \(=\) | \(\ds K_{18} + K_{19} + K_{20} + K_{21} + K_{22} + K_{23} + K_{24} + K_{25} + K_{26}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 15 \, 543 + 31 \, 053 + 62 \, 042 + 123 \, 961 + 247 \, 677 + 494 \, 865 + 988 \, 755 + 1 \, 975 \, 563 + 3 \, 947 \, 234\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7 \, 886 \, 693\) | ||||||||||||
\(\ds K_{28}\) | \(=\) | \(\ds K_{19} + K_{20} + K_{21} + K_{22} + K_{23} + K_{24} + K_{25} + K_{26} + K_{27}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 31 \, 053 + 62 \, 042 + 123 \, 961 + 247 \, 677 + 494 \, 865 + 988 \, 755 + 1 \, 975 \, 563 + 3 \, 947 \, 234 + 7 \, 886 \, 693\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 15 \, 757 \, 843\) | ||||||||||||
\(\ds K_{29}\) | \(=\) | \(\ds K_{20} + K_{21} + K_{22} + K_{23} + K_{24} + K_{25} + K_{26} + K_{27} + K_{28}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 62 \, 042 + 123 \, 961 + 247 \, 677 + 494 \, 865 + 988 \, 755 + 1 \, 975 \, 563 + 3 \, 947 \, 234 + 7 \, 886 \, 693 + 15 \, 757 \, 843\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 31 \, 484 \, 633\) | ||||||||||||
\(\ds K_{30}\) | \(=\) | \(\ds K_{21} + K_{22} + K_{23} + K_{24} + K_{25} + K_{26} + K_{27} + K_{28} + K_{29}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 123 \, 961 + 247 \, 677 + 494 \, 865 + 988 \, 755 + 1 \, 975 \, 563 + 3 \, 947 \, 234 + 7 \, 886 \, 693 + 15 \, 757 \, 843 + 31 \, 484 \, 633\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 62 \, 907 \, 224\) | ||||||||||||
\(\ds K_{31}\) | \(=\) | \(\ds K_{22} + K_{23} + K_{24} + K_{25} + K_{26} + K_{27} + K_{28} + K_{29} + K_{30}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 247 \, 677 + 494 \, 865 + 988 \, 755 + 1 \, 975 \, 563 + 3 \, 947 \, 234 + 7 \, 886 \, 693 + 15 \, 757 \, 843 + 31 \, 484 \, 633 + 62 \, 907 \, 224\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 125 \, 690 \, 487\) | ||||||||||||
\(\ds K_{32}\) | \(=\) | \(\ds K_{23} + K_{24} + K_{25} + K_{26} + K_{27} + K_{28} + K_{29} + K_{30} + K_{31}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 494 \, 865 + 988 \, 755 + 1 \, 975 \, 563 + 3 \, 947 \, 234 + 7 \, 886 \, 693 + 15 \, 757 \, 843 + 31 \, 484 \, 633 + 62 \, 907 \, 224 + 125 \, 690 \, 487\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 251 \, 133 \, 297\) |
Thus $251 \, 233 \, 297$ occurs in $K$ and the result follows by definition of Keith number.
$\blacksquare$
Historical Note
The Keith number $251 \, 133 \, 297$ was discovered, along with $129 \, 572 \, 008$, by Clifford A. Pickover, who reported it in his work Computers and the Imagination of $1991$.
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $129,572,008$