Sequences of 3 Consecutive Triangular Numbers which are Sphenic

Theorem
Let $n \in \N$ be a natural number such that:
 * $T_n$, $T_{n + 1}$ and $T_{n + 2}$ are all sphenic numbers

where $T_n$ denotes the $n$th triangular number.

The sequence of such $n$ begins:


 * $406$, $861$, $39 \, 621$, $2 \, 166 \, 321$, $3 \, 924 \, 201$, $11 \, 146 \, 281$, $14 \, 804 \, 961$, $19 \, 198 \, 306$, $73 \, 951 \, 041$, $83 \, 417 \, 986$, $97 \, 951 \, 006$, $209 \, 643 \, 526$, $310 \, 415 \, 986$, $522 \, 339 \, 681$, $526 \, 225 \, 461$, $583 \, 333 \, 246$, $611 \, 153 \, 241$, $801 \, 460 \, 666$, $1 \, 601 \, 581 \, 906$, $2 \, 520 \, 251 \, 506$, $2 \, 690 \, 954 \, 841$, $4 \, 455 \, 349 \, 606$, $6 \, 681 \, 853 \, 401$, $9 \, 895 \, 642 \, 221$, $13 \, 878 \, 029 \, 901$

Also see

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