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

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