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 |
$\blacksquare$