Infinite, Aperiodic Aliquot Sequences

 I've written about Aliquot Sequences in previous posts:

• Aliquot Sequences on 20th December 2017
• Aliquot Sequences Revisited on 21st June 2021
• The Lehmer Five on 17th May 2024

I was reminded of them again today as I turned 27528 days old. Running my multipurpose algorithm, I noticed that it stalled when calculating the aliquot sequence for 27528. On SageMathCell and on my Jupyter Notebook running on my laptop, I got to around 1000 steps without any termination. Using this site, I was able to check up to 2338 steps, still without termination. The final number at step 2338 was:

7025043146011116025148597113860868783698470017459754356208796140115478398029443564119146736882769530277894299741643314111035040751287025499046969087553315043196744

This of course can be factorised and the process continued but that was as far as the site was willing to go and a line has to be drawn somewhere and this was a reasonable place to stop I reckon.

27528 appears to generate an
infinite, aperiodic aliquot sequence

OEIS A131884 lists numbers up to 1836 that are conjectured to have infinite, aperiodic aliquot sequences.

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, 1512, 1560, 1572, 1578, 1590, 1632, 1650, 1662, 1674, 1722, 1734, 1758, 1770, 1806, 1836

27528 was not on the trajectory of any of these 45 numbers (constituting about 2.5% of the range). OEIS A216072 lists all numbers belonging to distinct families. These numbers are:

276, 552, 564, 660, 966, 1074, 1134, 1464, 1476, 1488, 1512, 1560, 1578, 1632, 1734, 1920, 1992, 2232, 2340, 2360, 2484, 2514, 2664, 2712, 2982, 3270, 3366, 3408, 3432, 3564, 3678, 3774, 3876, 3906, 4116, 4224, 4290, 4350, 4380, 4788, 4800, 4842

Notice the Lehmer Five numbers making their appearance. There are 81 numbers listed in all, up to 9852. I haven't tested all of these to see if 27528 is on one of their trajectories. Running my multipurpose algorithm nowadays as I do for every number associated with my diurnal age, I'll be able to detect any future numbers that have this same property that 27528 does.