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.