Primitive Abundant Numbers

Preliminary note: I've written about odd primitive abundant numbers in an earlier, eponymous post from May 21st 2017, so some content from that post is repeated here but there is new content as well. Here is the link.

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The sum of the proper divisors of an abundant number is greater than the number itself. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. So what is a primitive abundant number?

To quote from Numbers Aplenty:

An abundant number is called primitive if none of its proper divisors is abundant. 

There are infinitely many such numbers, both even and odd. However Dickson proved that there are only a finite number of odd primitive abundant numbers with a given number of distinct prime factors. 

For example, there are only 8 odd primitive abundant numbers with 3 distinct prime factors, namely, 945, 1575, 2205, 7425, 78975, 131625, 342225, and 570375. 

The first primitive abundant numbers are 12, 18, 20, 30, 42, 56, 66, 70, 78, 88, 102, 104, 114, 138, 174, 186, 196 more terms. 

A second definition of primitive numbers excludes also those that have perfect proper divisors, like all multiples of 6. The first such numbers are 20, 70, 88, 104, 272, 304, 368, 464, 550, 572, 650, 748, 836, 945, 1184, 1312, 1376, 1430, 1504, 1575, 1696, 1870, 1888, 1952, 2002.

Here are some properties of primitive abundant numbers taken from Wikipedia:

Every multiple of a primitive abundant number is an abundant number. 

Every abundant number is a multiple of a primitive abundant number or a multiple of a perfect number. 

Every primitive abundant number is either a primitive semiperfect (also called primitive pseudoperfect) number or a weird number. 

There are an infinite number of primitive abundant numbers. 

The number of primitive abundant numbers less than or equal to \(n\) is \( o \left( \frac{n}{\log^2(n)} \right)\ \). 

A semiperfect or pseudoperfect number is a natural number that is equal to the sum of all or some of its proper divisors. A primitive semiperfect number (also called a primitive pseudoperfect number, irreducible semiperfect number or irreducible pseudoperfect number) is a semiperfect number that has no semiperfect proper divisor. The first few primitive semiperfect numbers are 6, 20, 28, 88, 104, 272, 304, 350, ... There are infinitely many odd primitive semiperfect numbers, the smallest being 945.

A weird number is a natural number that is abundant but not semiperfect or pseudoperfect. In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself. The first few weird numbers are 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, ...

See my blog post titled Zumkellar, Half-Zumkellar, and Pseudoperfect Numbers and Odd Primitive Abundant Numbers.

Practical Numbers

Today I turned 25320 days old but I accidentally entered 25230 into the OEIS and discovered that it was a practical number, specifically one that formed the central member of a triple of practical numbers. It is a number \(n\) such that \(n-2\), \(n\), \(n+2\) are all practical numbers (OEIS A287682). So in this case, the other members of the triple are 25228 and 25232. This is not a common occurrence as can be seen by the initial members of the sequence:

4, 6, 18, 30, 198, 306, 462, 702, 1482, 2550, 3330, 4422, 5778, 6102, 6498, 9042, 11178, 11778, 14418, 15498, 17298, 17442, 19458, 20862, 21582, 22878, 23322, 23550, 25230, ...

This led me to investigate what characterised a practical number and during that process I realised that I'd wrongly entered my number of the day. However, as it turns out 24320 is also a practical number but not a member of a triplet (the next entry in OEIS A287682 is 26622). I was encouraged to continue my investigations. According to Wikipedia:

In number theory, a practical number or panarithmic number is a positive integer \(n\) such that all smaller positive integers can be represented as sums of distinct divisors of \(n\). For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors 1, 2, 3, 4, and 6: as well as these divisors themselves, we have 5 = 3 + 2, 7 = 6 + 1, 8 = 6 + 2, 9 = 6 + 3, 10 = 6 + 3 + 1, and 11 = 6 + 3 + 2.

The practical numbers themselves are rather frequent. OEIS A005153 lists these initial practical numbers:

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252, ...

As can be seen, all except 1 are multiples of 2 and this is one condition for a number being practical. The fully rigorous statement of what determines a practical number would be:

A positive integer greater than one with prime factorisation

\(n=p_1^{\alpha_1} \dots p_k^{\alpha_k}\) (with the primes in sorted order) is practical if and only if each of its prime factors \(p_i \) is small enough for \(p_i-1 \) to have a representation as a sum of smaller divisors. For this to be true, the first prime \(p_1 \) must equal 2 and, for every \(i \) from 2 to \( k \), each successive prime \(p_i \) must obey the inequality:$$ p_i\leq1+\sigma(p_1^{\alpha_1}p_2^{\alpha_2}\dots p_{i-1}^{\alpha_{i-1}})=1+\prod_{j=1}^{i-1}\frac{p_j^{\alpha_j+1}-1}{p_j-1}$$where \( \sigma(x) \) denotes the sum of the divisors of \(x\). For example, 2 × 3^2 × 29 × 823 = 429606 is practical, because the inequality above holds for each of its prime factors: 3 ≤ \(\ \sigma \)(2) + 1 = 4, 29 ≤ \( \sigma \)(2 × 3^2) + 1 = 40, and 823 ≤ \(\sigma \)(2 × 3^2 × 29) + 1 = 1171.

on August 1st 2021

With some difficulty I developed some SAGE code to calculate the practical numbers contained in a given range. The example at the end of this post is for the range from 26400 to 26500. It includes multiples of 2 because all powers of 2 are practical numbers along with all perfect numbers and primorials. Such numbers are not merely of interest to recreational mathematicians. They are of interest to professional mathematicians because many of their properties are similar to the properties of the prime numbers.

Deficient Numbers

Sometimes it's easy to forget the basics, such as what defines a deficient number. For example, today's number 25137 has the following entry in OEIS: deficient numbers n having a companion m > n such that sigma(n)/n = sigma(m)/m. The initial numbers in this sequence are shown below:

135, 3375, 1485, 2295, 2565, 3105, 3915, 4185, 4995, 5535, 5805, 6345, 25137, 7155, 7965, 8235, 9045, 9585, 9855, 10665, 11205, 12015, 13095, 13635, 13905, 14445, 14715, 43875, 15255, 16335, 17145, 17685, 18495, 18765, 57375, 20115, 20385, 21195, 64125

These numbers are listed in the order that their companions were found. All these numbers appear to have only one companion, which appear in A212609. The initial entries in this sequence are shown below with the 13th entry marked, namely 40131, because 25137 is the 13th entry in the previous set of numbers:

819, 6975, 9009, 13923, 15561, 18837, 23751, 25389, 30303, 33579, 35217, 38493, 40131, 43407, 48321, 49959, 54873, 58149, 59787, 64701, 67977, 72891, 79443, 82719, 84357, 87633, 89271, 90675, 92547, 99099, 104013, 107289, 112203, 113841, 118575, 122031

So the companion for 25137 is 40131 and checking we find that: $$ \frac {\sigma(25137)}{25137}=\frac{\sigma(40131)}{40131} \approx 1.81406 $$However, just to remind myself about the distinction between deficient, perfect and abundant numbers, I've included the following graphic:

By now the sigma function has begun to sink into my long term memory along with the Euler totient function or phi function as it's sometimes known.

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).

Odd Primitive Abundant Numbers

Today I turned 24885 days old. As usual I turned to WolframAlpha to find the prime number factorisation. It is \(3^2 \times 5 \times 7 \times 79\). I then turned to the Online Encyclopaedia of Integer Sequences (OEIS) to see what was special about the number. First mentioned was OEIS A006038: odd primitive abundant numbers. I was already familiar with abundant numbers. These are numbers in which the sum of the proper divisors exceeds the number itself. The first abundant number is 12 and the sum of its proper divisors (1, 2, 4 and 6) is 13. On the other hand, a number like 15 is called deficient because the sum of its proper divisors (1, 3 and 5) is only 9. A number like 6 is called perfect because the sum of its proper divisors (1, 2 and 3) is 6 and equals the number itself.

The divisors of 24885 turn out to be: 1 | 3 | 5 | 7 | 9 | 15 | 21 | 35 | 45 | 63 | 79 | 105 | 237 | 315 | 395 | 553 | 711 | 1185 | 1659 | 2765 | 3555 | 4977 | 8295 and total 25035. It is clearly an abundant number but what is a primitive abundant number? Well, Wikipedia supplies the following definition: in mathematics a primitive abundant number is an abundant number whose proper divisors are all deficient numbers. 20 is given an example of such a number because its divisors (1, 2, 4, 5 and 10) are all deficient numbers (the sums of the factors being respectively 0, 1, 3, 1 and 8). The sequence of primitive abundant numbers begins as follows (OEIS A091191):

12, 18, 20, 30, 42, 56, 66, 70, 78, 88, 102, 104, 114, 138, 174, 186, 196, 222, 246, 258, 272, 282, 304, 308, 318, 354, 364, 366, 368, 402, 426, 438, 464, 474, 476, 498, 532, 534, 550, 572, 582, 606, 618, 642, 644, 650, 654, 678, 748, 762, 786, 812, 822

It will be noted that any odd numbers are conspicuously absent from this initial list. As it turns out, the first odd primitive abundant number is 945. The sequence of odd primitive abundant numbers begins as follows (OEIS A006038):

945, 1575, 2205, 3465, 4095, 5355, 5775, 5985, 6435, 6825, 7245, 7425, 8085, 8415, 8925, 9135, 9555, 9765, 11655, 12705, 12915, 13545, 14805, 15015, 16695, 18585, 19215, 19635, 21105, 21945, 22365, 22995, 23205, 24885, 25935, 26145, 26565, 28035, 28215

So that's the story. There are:

• abundant numbers
• primitive abundant numbers
• even primitive abundant numbers
• odd primitive abundant numbers (far less frequent than their even counterparts)

Looking at the list of numbers of odd primitive abundant numbers, it can be seen that it will be about three years before I encounter another one.