There exist no 4 Consecutive Triangular Numbers which are all Sphenic Numbers

Theorem
Let $n \in \N$ be a natural number.

Let $T_n$, $T_{n + 1}$, $T_{n + 2}$, and $T_{n + 3}$ be the $n$th, $n + 1$th, $n + 2$th and $n + 3$th triangular numbers respectively.

Then it is not the case that all of $T_n$, $T_{n + 1}$, $T_{n + 2}$, and $T_{n + 3}$ are sphenic numbers.

Also see

 * Sequence of Smallest 3 Consecutive Triangular Numbers which are Sphenic: $T_{28}$, $T_{29}$ and $T_{30}$ that is, $406$, $435$ and $465$, are the smallest $3$ consecutive triangular numbers which are all sphenic.


 * Sequences of 3 Consecutive Triangular Numbers which are Sphenic