Numbers with 6 or more Prime Factors

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Theorem

The sequence of positive integers with $6$ or more prime factors (not necessarily distinct) begins:

$64, 96, 128, 144, 160, 192, 216, 224, 240, 256, \ldots$

This sequence is A046305 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

\(\ds 64\) \(=\) \(\ds 2 \times 2 \times 2 \times 2 \times 2 \times 2\)
\(\ds 96\) \(=\) \(\ds 2 \times 2 \times 2 \times 2 \times 2 \times 3\)
\(\ds 128\) \(=\) \(\ds 2 \times 2 \times 2 \times 2 \times 2 \times 2 \paren {\times 2}\)
\(\ds 144\) \(=\) \(\ds 2 \times 2 \times 2 \times 2 \times 3 \times 3\)
\(\ds 160\) \(=\) \(\ds 2 \times 2 \times 2 \times 2 \times 2 \times 5\)
\(\ds 192\) \(=\) \(\ds 2 \times 2 \times 2 \times 2 \times 2 \times 2 \paren {\times 3}\)
\(\ds 216\) \(=\) \(\ds 2 \times 2 \times 2 \times 3 \times 3 \times 3\)
\(\ds 224\) \(=\) \(\ds 2 \times 2 \times 2 \times 2 \times 2 \times 7\)
\(\ds 240\) \(=\) \(\ds 2 \times 2 \times 2 \times 2 \times 3 \times 5\)
\(\ds 256\) \(=\) \(\ds 2 \times 2 \times 2 \times 2 \times 2 \times 2 \paren {\times 2 \times 2}\)

$\blacksquare$


Sources