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.

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.

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.