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
\(\ds 128\) | \(=\) | \(\ds 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\) | ||||||||||||
\(\ds 192\) | \(=\) | \(\ds 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3\) | ||||||||||||
\(\ds 256\) | \(=\) | \(\ds 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \left({\times 2}\right)\) | ||||||||||||
\(\ds 288\) | \(=\) | \(\ds 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3\) | ||||||||||||
\(\ds 320\) | \(=\) | \(\ds 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5\) | ||||||||||||
\(\ds 384\) | \(=\) | \(\ds 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \left({\times 3}\right)\) | ||||||||||||
\(\ds 432\) | \(=\) | \(\ds 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3\) | ||||||||||||
\(\ds 448\) | \(=\) | \(\ds 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 7\) | ||||||||||||
\(\ds 480\) | \(=\) | \(\ds 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 5\) | ||||||||||||
\(\ds 512\) | \(=\) | \(\ds 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \left({\times 2 \times 2}\right)\) |
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $128$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $128$