Sequence of Smallest 3 Consecutive Triangular Numbers which are Sphenic

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Theorem

The smallest $3$ consecutive triangular numbers which are sphenic is:

$406$, $435$, $465$


Proof

Let $T_n$ denote the $n$th triangular number.

The smallest sphenic number is $30$.

Hence we need investigate triangular number from where $T_n \ge 30$.

Thus:

\(\ds T_8\) \(=\) \(\ds 36\) \(\ds = 2^2 \times 3^2\) which is not sphenic
\(\ds T_9\) \(=\) \(\ds 45\) \(\ds = 3^2 \times 5\) which is not sphenic
\(\ds T_{10}\) \(=\) \(\ds 55\) \(\ds = 5 \times 11\) which is not sphenic
\(\ds T_{11}\) \(=\) \(\ds 66\) \(\ds = 2 \times 3 \times 11\) which is sphenic
\(\ds T_{12}\) \(=\) \(\ds 78\) \(\ds = 2 \times 3 \times 13\) which is sphenic
\(\ds T_{13}\) \(=\) \(\ds 91\) \(\ds = 7 \times 13\) which is not sphenic
\(\ds T_{14}\) \(=\) \(\ds 105\) \(\ds = 3 \times 5 \times 7\) which is sphenic
\(\ds T_{15}\) \(=\) \(\ds 120\) \(\ds = 2^2 \times 3 \times 5\) which is not sphenic
\(\ds T_{16}\) \(=\) \(\ds 135\) \(\ds = 2^3 \times 17\) which is not sphenic
\(\ds T_{17}\) \(=\) \(\ds 153\) \(\ds = 3^2 \times 17\) which is not sphenic
\(\ds T_{18}\) \(=\) \(\ds 171\) \(\ds = 3^2 \times 19\) which is not sphenic
\(\ds T_{19}\) \(=\) \(\ds 190\) \(\ds = 2 \times 5 \times 19\) which is sphenic
\(\ds T_{20}\) \(=\) \(\ds 210\) \(\ds = 2 \times 3 \times 5 \times 7\) which is not sphenic
\(\ds T_{21}\) \(=\) \(\ds 231\) \(\ds = 3 \times 7 \times 11\) which is sphenic
\(\ds T_{22}\) \(=\) \(\ds 253\) \(\ds = 11 \times 23\) which is not sphenic
\(\ds T_{23}\) \(=\) \(\ds 276\) \(\ds = 2^2 \times 3 \times 23\) which is not sphenic
\(\ds T_{24}\) \(=\) \(\ds 300\) \(\ds = 2^2 \times 3 \times 5^2\) which is not sphenic
\(\ds T_{25}\) \(=\) \(\ds 325\) \(\ds = 5^2 \times 13\) which is not sphenic
\(\ds T_{26}\) \(=\) \(\ds 351\) \(\ds = 3^3 \times 13\) which is not sphenic
\(\ds T_{27}\) \(=\) \(\ds 351\) \(\ds = 2 \times 3^3 \times 7\) which is not sphenic
\(\ds T_{28}\) \(=\) \(\ds 406\) \(\ds = 2 \times 7 \times 29\) which is sphenic
\(\ds T_{29}\) \(=\) \(\ds 435\) \(\ds = 3 \times 5 \times 29\) which is sphenic
\(\ds T_{30}\) \(=\) \(\ds 465\) \(\ds = 3 \times 5 \times 31\) which is sphenic

$\blacksquare$


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