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
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $64$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $96$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $64$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $96$