Divisors and Antidivisors: A Fresh Perspective

I've written about antidivisors before in posts titled Anti-Divisors and More on Anti-Divisors. Here is a fresh take. My diurnal age today is 28134 and the antidivisors of this number are 4, 12, 36, 108, 2084, 6252 and 18756. If we concatenate these numbers from left to right we get a rather large number (412361082084625218756) with the property that:$$ \frac{412361082084625218756}{28134} = 14657037111133334$$Numbers with this property form OEIS A249764:

 

A249764: numbers which divide the concatenation, in ascending order, of their anti-divisors.

Up to 40000, these numbers are relatively rare (permalink): 

15, 30, 105, 120, 150, 222, 375, 585, 1500, 1695, 1755, 1800, 2700, 3449, 3750, 3840, 4891, 6720, 7680, 12000, 13583, 14400, 15000, 18750, 19200, 20940, 28134, 30000, 34800, 35625

The OEIS lists an analog of this, namely OEIS A069872:

 

A069872: numbers \(k\) such that \(k\) divides the concatenation all divisors in ascending order.

Comparatively, members of this sequence are rather more numerous. There are 181 numbers in the range up to 40000 (permalink):

1, 2, 4, 5, 6, 8, 10, 15, 16, 20, 24, 25, 30, 32, 40, 50, 60, 64, 80, 90, 96, 100, 104, 120, 124, 125, 128, 150, 160, 200, 240, 250, 255, 256, 288, 320, 360, 375, 380, 384, 400, 425, 464, 480, 495, 500, 512, 600, 618, 625, 640, 750, 795, 800, 864, 875, 960, 1000, 1024, 1110, 1230, 1250, 1280, 1300, 1390, 1400, 1408, 1440, 1469, 1500, 1525, 1536, 1600, 1632, 1920, 2000, 2048, 2050, 2250, 2400, 2500, 2556, 2560, 2910, 2944, 2952, 3000, 3040, 3125, 3200, 3330, 3360, 3625, 3750, 3825, 3840, 4000, 4096, 4304, 4625, 4650, 4992, 5000, 5120, 5250, 5280, 5300, 5568, 5760, 6000, 6144, 6150, 6250, 6400, 6528, 6750, 7168, 7200, 7560, 7680, 8000, 8192, 9000, 9330, 9375, 9600, 10000, 10240, 10500, 10752, 11875, 12000, 12500, 12800, 13410, 13600, 14000, 15000, 15360, 15625, 15680, 16000, 16384, 16640, 17500, 17920, 18432, 18750, 19710, 20000, 20200, 20250, 20480, 20610, 21600, 21760, 22752, 23040, 23375, 24000, 24120, 24576, 25000, 25600, 25984, 27000, 28125, 30720, 31250, 32000, 32768, 33930, 34480, 34800, 35000, 36000, 37500, 37830, 38400, 39000, 40000

A variation on this is OEIS A240265:

A240265: numbers that divide the concatenation of their aliquot divisors, in ascending order.

These are less numerous. In the range up to 40000, they are:

4, 15, 16, 255, 375, 495, 795, 1469, 3825, 9375, 28125

All these numbers are also in OEIS A069872 and are marked in red in that sequence's list of members. It can be seen that 28125 is rather special and this number marked my diurnal age only 9 days ago. However, at the time, I missed its membership in these two sequences.