Consider those natural numbers $n$ such that $7\times10^n+69$ is a prime number. The first $15$ such numbers are $1$, $2$, $3$, $6$, $7$, $8$, $10$, $12$, $13$, $21$, $46$, $68$, $91$, $153$, and $366$. I would say that this is a particulary uninteresting sequence (by the way, this is sequence A294484 in OEIS).
However, it suddenly became interesting to me when I noticed that, for each number $n$ on this sequence, the lights-out problem has a unique solution on a $n\times n$ square board. The numbers for which this property holds form the sequence A076436 in OEIS. The two sequences are not equal; for instance, $15$, $18$, and $20$ belong to the second sequence, but not to the first one. But could it be that every element of the first sequence also belongs to the second one?
This is only a partial answer.
Could this be, at least in part, due to different phenomena having a common cause? I notice that, as far as it goes, A076436 contains no $n$ where $n=4\bmod 5$. $n=4\bmod 5\implies 41\mid 7\cdot10^n+69\implies n\not\in A294484$.