Aliquot Sequences

My attention was drawn to aliquot sequences today, day 25098, because the number is a member of four aliquot sequences (as shown below):

From any starting point, it's easy enough to calculate the next member in the sequence by using the divisor function \(\sigma_1 \). For example, the second term in the sequence starting with 138 is \(\sigma_1 (138) -138=150\). Many sequences lead to a prime number and then terminate because \(\sigma_1 (\text{prime number}) -\text{prime number}=1\) and \(\sigma_1(1) -1=0\). The sequence beginning with 138 (OEIS A008888) has 178 members and ends in 59, 1, 0. OEIS A008889 is really the same as OEIS A008888 except for the starting point (150 instead of 138)

Aliquot sequence OEIS A008890 is different however, and starts with 168 but the second term is 312 which is the fifth term in OEIS A008888. Aliquot sequence OEIS A074907 starts with 570 but after a few terms reaches 19434 which again is a term in the OEIS A08888 sequence.

Not all aliquot sequences end. To quote from Wikipedia:

There are a variety of ways in which an aliquot sequence might not terminate:

• A perfect number has a repeating aliquot sequence of period 1. The aliquot sequence of 6, for example, is 6, 6, 6, 6, ... 

• An amicable number has a repeating aliquot sequence of period 2. For instance, the aliquot sequence of 220 is 220, 284, 220, 284, ... 

• A sociable number has a repeating aliquot sequence of period 3 or greater. (Sometimes the term sociable number is used to encompass amicable numbers as well.) For instance, the aliquot sequence of 1264460 is 1264460, 1547860, 1727636, 1305184, 1264460, ... 

• Some numbers have an aliquot sequence which is eventually periodic, but the number itself is not perfect, amicable, or sociable. For instance, the aliquot sequence of 95 is 95, 25, 6, 6, 6, 6, ... . Numbers like 95 that are not perfect, but have an eventually repeating aliquot sequence of period 1 are called aspiring numbers (OEIS  A063769).

Numbers whose Aliquot sequence is not known to be finite or eventually periodic are:

276, 306, 396, 552, 564, 660, 696, 780, 828, 888, 966, 996, 1074, 1086, 1098, 1104, 1134, 1218, 1302, 1314, 1320, 1338, 1350, 1356, 1392, 1398, 1410, 1464, 1476, 1488, ... (sequence A131884 in the OEIS) 

ADDENDUM: 17th July 2020

Today, I turned 26038 days old and I revisited OEIS A008888 (Aliquot sequence starting at 138) because this number is a member of the sequence and appears very near the end. Here is the full sequence:

138, 150, 222, 234, 312, 528, 960, 2088, 3762, 5598, 6570, 10746, 13254, 13830, 19434, 20886, 21606, 25098, 26742, 26754, 40446, 63234, 77406, 110754, 171486, 253458, 295740, 647748, 1077612, 1467588, 1956812, 2109796, 1889486, 953914, 668966, 353578, 176792, 254128, 308832, 502104, 753216, 1240176, 2422288, 2697920, 3727264, 3655076, 2760844, 2100740, 2310856, 2455544, 3212776, 3751064, 3282196, 2723020, 3035684, 2299240, 2988440, 5297320, 8325080, 11222920, 15359480, 19199440, 28875608, 25266172, 19406148, 26552604, 40541052, 54202884, 72270540, 147793668, 228408732, 348957876, 508132204, 404465636, 303708376, 290504024, 312058216, 294959384, 290622016, 286081174, 151737434, 75868720, 108199856, 101437396, 76247552, 76099654, 42387146, 21679318, 12752594, 7278382, 3660794, 1855066, 927536, 932464, 1013592, 1546008, 2425752, 5084088, 8436192, 13709064, 20563656, 33082104, 57142536, 99483384, 245978376, 487384824, 745600776, 1118401224, 1677601896, 2538372504, 4119772776, 8030724504, 14097017496, 21148436904, 40381357656, 60572036544, 100039354704, 179931895322, 94685963278, 51399021218, 28358080762, 18046051430, 17396081338, 8698040672, 8426226964, 6319670230, 5422685354, 3217383766, 1739126474, 996366646, 636221402, 318217798, 195756362, 101900794, 54202694, 49799866, 24930374, 17971642, 11130830, 8904682, 4913018, 3126502, 1574810, 1473382, 736694, 541162, 312470, 249994, 127286, 69898, 34952, 34708, 26038, 13994, 7000, 11720, 14740, 19532, 16588, 18692, 14026, 7016, 6154, 3674, 2374, 1190, 1402, 704, 820, 944, 916, 694, 350, 394, 200, 265, 59, 1, 0.

This is not the last time I'll encounter OEIS A008888 because in a couple of years time, I'll meet 26742 and 26754 (assuming I'm still alive).