Numbers with 7 or more Prime Factors

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Theorem

The sequence of positive integers with $7$ or more prime factors (not necessarily distinct) begins:

$128, 192, 256, 288, 320, 384, 432, 448, 480, 512, \ldots$

This sequence is A046307 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

\(\displaystyle 128\) \(=\) \(\displaystyle 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\)
\(\displaystyle 192\) \(=\) \(\displaystyle 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3\)
\(\displaystyle 256\) \(=\) \(\displaystyle 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \left({\times 2}\right)\)
\(\displaystyle 288\) \(=\) \(\displaystyle 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3\)
\(\displaystyle 320\) \(=\) \(\displaystyle 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5\)
\(\displaystyle 384\) \(=\) \(\displaystyle 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \left({\times 3}\right)\)
\(\displaystyle 432\) \(=\) \(\displaystyle 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3\)
\(\displaystyle 448\) \(=\) \(\displaystyle 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 7\)
\(\displaystyle 480\) \(=\) \(\displaystyle 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 5\)
\(\displaystyle 512\) \(=\) \(\displaystyle 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \left({\times 2 \times 2}\right)\)

$\blacksquare$


Sources