Numbers with 7 or more Prime Factors

From ProofWiki
Jump to navigation Jump to search

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

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

$\blacksquare$


Sources