Aliquant Parts

I came across a mathematical term today that I hadn't heard of before. The term is "aliquant" defined as follows by contrasting it to similar sounding "aliquot" (source):

Webster defines 'aliquot' as something that contained an exact number of times in something else or to divide into equal parts.

Notice the word "equal". An example being 5 is an aliquot part of 15.

The term 'aliquant', however, is slightly different. Defined as being a part of a number or quantity, but not dividing it without leaving a remainder. An example being 5 is an aliquant part of 16. 

The term occurred in the following context:

 A098743: number of partitions of \(n\) into aliquant parts (i.e., parts that do not divide \(n\)). 

The initial members of the sequence are:

1, 0, 0, 0, 0, 1, 0, 3, 1, 3, 3, 13, 1, 23, 10, 11, 9, 65, 8, 104, 14, 56, 66, 252, 10, 245, 147, 206, 77, 846, 35, 1237, 166, 649, 634, 1078, 60, 3659, 1244, 1850, 236, 7244, 299, 10086, 1228, 1858, 4421, 19195, 243, 17660, 3244, 12268, 4039, 48341, 1819, 27675

The last member shown above, 27675, is my diurnal age today and corresponds to \(n=55\). To find the aliquant parts, simply remove the divisors of the number. For example, 55 has divisors of 1, 5, 11 and 55 and so the number of aliquant parts is 51. The list is as follows:

2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54

Note that these are different to the numbers that contribute to the total of the totient of a number, where the numbers are coprime. The totient for 55 is 40, made up of the following numbers.

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 16, 17, 18, 19, 21, 23, 24, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 39, 41, 42, 43, 46, 47, 48, 49, 51, 52, 53, 54

Up and Down the Mountain

The number (27532) associated with my diurnal age today has an interesting aliquot sequence consisting of 210 steps that reaches a maximum value of 210998991785527991104 before beginning its descent to 1 and then 0. Here is the sequence:

27532, 20656, 19396, 17256, 25944, 43176, 80664, 121056, 224688, 378448, 494512, 495504, 1012336, 1181968, 1182960, 2995344, 6599280, 14542224, 25693296, 43014360, 90683160, 185451240, 425275800, 940708200, 1975489080, 4299600360, 9787608600, 30598377960, 62464790040, 124929580440, 322138579560, 782543654040, 1590997194600, 3789466067640, 8612422885320, 17245450814520, 34491361043400, 72967727801400, 156608347258440, 511373352229560, 1193233352540040, 3238776242618040, 7557144566112360, 17008403414134680, 43343078672652120, 98245289952891720, 225485753448117420, 499595022460258740, 1016220356878620060, 2146302511659329220, 4365805465306640340, 8912079582213674700, 19057572060429045492, 30451944343316954508, 46523803857845347256, 43522269061507054024, 41538397982866392056, 39514226617193027944, 34588280875055182556, 26164545907884419524, 19879736397904788476, 15278939116317231844, 13034324430483320084, 9787165489989868000, 14259900118915247504, 13373492843095311856, 12537649540401854896, 11754756084394941888, 20702681104040261952, 35918311240577926848, 63312611494171175232, 104258325352849123008, 210998991785527991104, 207702132538879116370, 169624609562080576430, 135726953604303765970, 148468761008721086318, 131290689970046993554, 93931702661206931822, 67094073456072645490, 63039859860977978510, 66642137567319577426, 33622680034706422574, 20453938384323035986, 10514105806833405614, 5257053195212561074, 2628558999616292174, 1319289459782004154, 678859735840149446, 432198199089642970, 372584654387623790, 423185411483397010, 447367434996734126, 335199506688516274, 241133173013919566, 129392986488928594, 69210234498425006, 49442455265686354, 31628275705499822, 17569420012749490, 15482802556130510, 12912073075113586, 8615431262813582, 4307715631406794, 2182074293639066, 1091037352376614, 545519846484554, 272807528144026, 136403764072016, 172028060840272, 246230032928432, 322698987259984, 302536923800196, 440536222376284, 330402166782220, 469768933828340, 521929534249180, 574124185108340, 632252256474700, 811995150775220, 893194665852784, 856062512768896, 913869490270304, 885311068699420, 1142969402332388, 857227051749298, 428613525874652, 321536752958884, 279499971575516, 235369388238244, 208212125817436, 207269989503284, 155452492127470, 125699432957330, 100559546365882, 50279927860454, 25488855451066, 20776630073990, 21985463380090, 25902201396614, 14856456204922, 7434249373850, 7457427003070, 7186247839538, 4160459275582, 2399770256450, 2707938109054, 1364410976426, 711333048214, 360723394754, 183945042814, 115353159818, 74577627382, 39307973018, 20226433030, 16181146442, 8167141114, 5307725702, 2873228938, 1532896886, 800767234, 435474206, 217915858, 109021742, 58683250, 51172262, 26101738, 16062650, 15625798, 8569082, 5026822, 2524250, 2417830, 1934282, 1381654, 746954, 459706, 282938, 144250, 126254, 63130, 53510, 42826, 39254, 22786, 11396, 14140, 20132, 20188, 21308, 21364, 22526, 16114, 11534, 6226, 3998, 2002, 2030, 2290, 1850, 1684, 1270, 1034, 694, 350, 394, 200, 265, 59, 1, 0

Figure 1 shows these values plotted on a logarithmic vertical scale that necessarily ends in 1 not 0 because we are working with logarithms. The sequence is embedded in the graph.

Figure 1: permalink (not annotated)

The logarithmic graph is preferable to the non-logarithmic because of the massive spike that makes the smaller values invisible. See Figure 2.

Figure 2: permalink

In an earlier post the 15th August 2024, I posted about Infinite, Aperiodic Aliquot Series but this series is finite as shown but takes a while to terminate. I'll continue to monitor the aliquot sequences generated by the numbers associated with my diurnal age and report on any that involve a large number of steps to terminate or for which no termination can be demonstrated.

In earlier posts, I've mentioned aliquot sequences of various types. These posts include:

• Aliquot Sequences on 20th December 2017
• Aliquot Sequences Revisited on 21st June 2021
• Periodic Unitary Aliquot Sequences on 30th September 2023
• The Lehmer Five on 17th May 2024
• Infinite, Aperiodic Aiquot Sequences on 15th August 2023

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.

The Lehmer Five

276, 552, 564, 660, 966

***********************************

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

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.

Untouchable Numbers

I’ve dealt with untouchable numbers before but only in passing. I’ve never devoted an entire post to the topic. In my Mathematical Meandering blog on Blogger, I mentioned this category of numbers in two posts: one titled The Connectivity of Numbers and the other Mathematical Properties of 2022. My diurnal age today happens to be 26708 and this number turns out to be untouchable, meaning that there are no numbers whose sum of aliquot parts is equal to this number.

26798 is the 3470th untouchable number, meaning that the frequency of such numbers over this range is about 13%. As the range is extended, it has been shown that this density remains at least greater than 6% and Paul Erdos has shown that there are infinitely many untouchable numbers. These numbers are listed in OEIS A005114 and the sequence begins:

2, 5, 52, 88, 96, 120, 124, 146, 162, 188, 206, 210, 216, 238, 246, 248, 262, 268, 276, 288, 290, 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, 516, 518, 520, 530, 540, 552, 556, 562, 576, 584, 612, 624, 626, 628, 658, …

It should be noted that a number cannot be untouchable if it is one more than a prime number \( p \) because then it would be the sum of the aliquot parts of \( p^2 \). Similarly, if a number is three more a prime number \( p \), it cannot be untouchable because then it would be the aliquot sum of \( 2p \). More formally, we can say that untouchable numbers are those numbers that are not in the range of the aliquot sum function \( s(n) \) where \( n \) is any positive integer and \( d \) represents its divisors:$$s(n)=\sum_{d|n, d \neq n} \! \! \! d $$The conjecture is that 5 is the only odd untouchable number but this has not been proven. If true, then all untouchable numbers are composite except 2. Writing a program to generate untouchable numbers is not as simple as it might seem because some quite large numbers can have a relatively small aliquot sum. Below is an SageMath algorithm that will generate the untouchable numbers in the range from 26798 to 26750 (permalink). If the search range drops much below 500,000, “false positives” will begin to appear. Feel free to experiment.