Count Down Number Chains

It's hard to decide on a suitable name for this class of numbers but let's recall the process by which I discovered them. Yesterday I turned 27286 days old and it was the factorisation of this number that caught my attention. $$ \begin{align} 27286 &= 2 \times 7 \times 1949\\ &=14 \times 1949 \end{align}$$As can be seen, the factorisation involved the number 1949 which is my year of birth. Interesting enough but it was the number associated with my diurnal age today, 27287, that got me thinking:$$27287 = 13 \times 2099$$Naturally I wondered if there was a pattern here and indeed there was. To cut to the chase, I discovered the following pattern:$$ \begin{align} 27285 &= 15 \times 1855\\27286 &= 14 \times 1949 \\27287 &= 13 \times 2099\\27288 &= 12 \times 2274 \end{align}$$The sequence is contained within 27284 that factorises to 2 x 2 x 19 x 359 and 27289 that factorises to 29 x 941. The next question that occurred to me was how common is such a sequence. It turns out that it is not so common. In the range of up to 40,000, there are only seven such sequences:$$5445 , 5446 , 5447 , 5448\\10905 , 10906 , 10907 , 10908\\16365 , 16366 , 16367 , 16368\\21825 , 21826 , 21827 , 21828\\27285 , 27286 , 27287 , 27288\\32745 , 32746 , 32747 , 32748\\38205 , 38206 , 38207 , 38208$$There is only one number chain that starts with 16 and goes down to 12 (in the range up to 40,000) and that is as follows (permalink):$$ \begin{align} 21824 &=16 \times 1364\\21825 &= 15 \times 1455\\21826 &=14 \times 1559\\21827 &= 13 \times 1679\\21828 &= 12 \times 1819 \end{align}$$Interestingly, if we start with 12 and count down to 6, we get a number chain that is seven terms long (in the range up to 40,000):$$ \begin{align} 27708 &= 12 \times 2309\\27709 &= 11 \times 2519\\27710 &= 10 \times 2771\\27711 &= 9 \times 3079\\27712 &= 8 \times 3464\\27713 &= 7 \times 3959\\27714 &= 6 \times 4619 \\27715 &= 5 \times 5543 \end{align}$$There's obviously more room for research and discovery regarding this topic but this is at least a small start.