Triangular Numbers which are Product of 3 Consecutive Integers

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Theorem

The $6$ triangular numbers which can be expressed as the product of $3$ consecutive integers are:

$6, 120, 210, 990, 185 \, 836, 258 \, 474 \, 216$

This sequence is A001219 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

\(\displaystyle T_3\) \(=\) \(\displaystyle \frac {3 \left({3 + 1}\right)} 2\) Closed Form for Triangular Numbers
\(\displaystyle \) \(=\) \(\displaystyle 6\)
\(\displaystyle \) \(=\) \(\displaystyle 1 \times 2 \times 3\)


\(\displaystyle T_{15}\) \(=\) \(\displaystyle \frac {15 \left({15 + 1}\right)} 2\) Closed Form for Triangular Numbers
\(\displaystyle \) \(=\) \(\displaystyle 120\)
\(\displaystyle \) \(=\) \(\displaystyle 4 \times 5 \times 6\)


\(\displaystyle T_{20}\) \(=\) \(\displaystyle \frac {20 \left({20 + 1}\right)} 2\) Closed Form for Triangular Numbers
\(\displaystyle \) \(=\) \(\displaystyle 210\)
\(\displaystyle \) \(=\) \(\displaystyle 5 \times 6 \times 7\)


\(\displaystyle T_{44}\) \(=\) \(\displaystyle \frac {44 \left({44 + 1}\right)} 2\) Closed Form for Triangular Numbers
\(\displaystyle \) \(=\) \(\displaystyle 990\)
\(\displaystyle \) \(=\) \(\displaystyle 9 \times 10 \times 11\)


\(\displaystyle T_{608}\) \(=\) \(\displaystyle \frac {608 \left({608 + 1}\right)} 2\) Closed Form for Triangular Numbers
\(\displaystyle \) \(=\) \(\displaystyle 185 \, 136\)
\(\displaystyle \) \(=\) \(\displaystyle 56 \times 57 \times 58\)


\(\displaystyle T_{22 \, 736}\) \(=\) \(\displaystyle \frac {22 \, 736 \left({22 \, 736 + 1}\right)} 2\) Closed Form for Triangular Numbers
\(\displaystyle \) \(=\) \(\displaystyle 258 \, 474 \, 216\)
\(\displaystyle \) \(=\) \(\displaystyle 2^3 \times 3 \times 7^2 \times 11 \times 13 \times 29 \times 53\)
\(\displaystyle \) \(=\) \(\displaystyle \left({2^2 \times 3 \times 53}\right) \times \left({7^2 \times 13}\right) \times \left({2 \times 11 \times 29}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle 636 \times 637 \times 638\)



Sources