Periodic Unitary Aliquot Sequences.

At first glance, the phrase "periodic unitary aliquot sequences" can sound intimidating so it needs to be broken down into its individual components. Let's start with a definition of aliquot taken from study.com:

An aliquot is a portion or part of a larger whole. An aliquot, or the aliquot part as it is referred to in mathematics, is defined as a positive proper divisor of a number. A divisor refers to a whole number that can be divided evenly into a number.

Using the number associated with my diurnal age today, 27208, the aliquot parts of this number are 1, 2, 4, 8, 19, 38, 76, 152, 179, 358, 716, 1432, 3401, 6802 and 13604. 

The next term to deal with is unitary. The unitary divisors of a number are defined by Wikipedia as follows:

\(a\) is a unitary divisor (or Hall divisor) of a number \(b\) if \(a\) is a divisor of \(b\) and if \(a\) and \(b/a\) are coprime, having no common factor other than 1. Thus, 5 is a unitary divisor of 60, because 5 and 60/5 =12 have only 1 as a common factor, while 6 is a divisor but not a unitary divisor of 60, as 6 and 60/6 = 10 have a common factor other than 1, namely 2. 1 is a unitary divisor of every natural number. 

In the case of 27208, the unitary divisors are 1, 8, 19, 152, 179, 1432, 3401, 27208 but the unitary aliquot divisors are 1, 8, 19, 152, 179, 1432, 3401. See my blog post Unitary Divisors.

Next we need to tackle aliquot sequences. Here is a definition:

Now in the case of 27208, the aliquot sequence terminates in zero. Here is the trajectory (permalink):

27208, 26792, 26668, 21212, 15916, 13316, 9994, 5846, 3274, 1640, 2140, 2396, 1804, 1724, 1300, 1738, 1142, 574, 434, 334, 170, 154, 134, 70, 74, 40, 50, 43, 1, 0

This sequence is terminating and not periodic. However, a unitary aliquot sequence uses the unitary divisors and can behave quite differently. In the case of 27208, the sequence becomes periodic with the following trajectory (permalink):

27208, 5192, 1288, 440, 208, 30, 42, 54, 30

27208 is a member of OEIS A003062:

A003062

Beginnings of periodic unitary aliquot sequences.

The initial members are:

6, 30, 42, 54, 60, 66, 78, 90, 100, 102, 114, 126, 140, 148, 194, 196, 208, 220, 238, 244, 252, 274, 288, 292, 300, 336, 348, 350, 364, 374, 380, 382, 386, 388, 400, 420, 436, 440, 476, 482, 484, 492, 516, 528, 540, 542, 550, 570, 578, 592, 600, 612, 648, 660, 680, 688, 694, 708, 720, 722, 740, 756, 758, 764, 766, 770, 780, 784, 792, 794, 812

Thus periodicity is fairly common for these types of sequences, quite unlike the aliquot sequences which nearly all end in 0. See my post Aliquot Sequences.