The Lehmer Five

276, 552, 564, 660, 966

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On the 20th November 2021, I created a post titled 888 in which I mentioned one of the properties of that number being that it is on the trajectory of 552:

A014360

Aliquot sequence starting at 552.

The sequence begins: 

552, 888, 1392, 2328, 3552, 6024, 9096, 13704, 20616, 30984

To quote from Wolfram Alpha:

It has not been proven that all aliquot sequences eventually terminate and become periodic. The smallest number whose fate is not known is 276. There are five such sequences less than 1000, namely 276, 552, 564, 660, and 966, sometimes called the "Lehmer five". 

I was reminded of the Lehmer five again thanks to one of the properties associated with my diurnal age today (which is 27438):

A014363

Aliquot sequence starting at 966.

The sequence runs:

966, 1338, 1350, 2370, 3390, 4818, 5838, 7602, 9870, 17778, 17790, 24978, 27438, 30882, 30894, 34386, 40782, 52530, 82254, 82266, 82278, 121770, 241110, 450090, 750870, 1295226, 1572678, 1919538, 2760984, 4964136

This particular site will track the aliquot trajectory of any given number to over a thousand terms if needed, providing factorisations for each term. Figure 1 shows a graph of 966 in terms of the number of digits in each term (rather than the actual value of each term) plotted against position in the sequence for the first thousand or so terms.

Figure 1: source

The Lehmer five mark the first five terms in OEIS A216072:

A216072

Aliquot open end sequences which belong to distinct families.

The initial terms 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

The OEIS comments state that:

These aliquot sequences are believed to grow forever without terminating in a prime or entering a cycle. Sequence A131884 lists all the starting values of an aliquot sequence that lead to open-ending. It includes all values obtained by iterating from the starting values of this sequence. But this sequence lists only the values that are the lowest starting elements of open end aliquot sequences that are the part of different open-ending families. 

V. Raman, Dec 08 2012