Zumkellar Numbers, Half Zumkellar Numbers and Pseudoperfect Numbers

Zumkeller numbers are related to perfect numbers such as 6, the divisors of which can be written as the set {1, 2, 3, 6}. There are two mutually exclusive subsets of this set, {1, 2, 3}  and {6}, whose union is the original set and both of whose elements add to 6. Zumkeller numbers are similar in that there are two mutually exclusive subsets of this set of divisors, whose union is the original set but with the difference that both of the elements in each subset add to a number other than the originating number. For example, 20 has the set of divisors {1, 2, 4, 5, 10, 20} and there are two subsets {1, 20} and {2, 4, 5, 10} that both total 21. Thus 20 is a Zumkellar number. Of course \( \sigma(20) \), the sum of the divisors of 20, is 42 and so each subset sums to half of that.

If \( n \) is a Zumkeller number, then \( \sigma(n) \) is even and \(n\) is perfect or abundant. A number is abundant if the sum of its proper divisors is greater than the number. All the practical numbers \( n \), with \( \sigma(n) \) even, are also Zumkeller numbers. Here is a link to my blog post on practical numbers.

Bhakara Rao & Peng have proved several results on Zumkeller numbers such as the fact that \(n!\) is a Zumkeller numbers for \(n\ge 3 \). For example, 5! =120 and the set of divisors is {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120}. This set can be divided into {60, 120} and {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40}, both of which total 180. For what it's worth, the following YouTube video features Bhakara Rao introducing the paper that he wrote with Peng.

OEIS A083207 lists the initial Zumkeller numbers:

6, 12, 20, 24, 28, 30, 40, 42, 48, 54, 56, 60, 66, 70, 78, 80, 84, 88, 90, 96, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 150, 156, 160, 168, 174, 176, 180, 186, 192, 198, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260, 264, 270, 272, ...

In the OEIS comments for this sequence, it's stated that:

The 229026 Zumkeller numbers less than one million have a maximum difference of 12. This leads to the conjecture that any 12 consecutive numbers include at least one Zumkeller number. There are 1989 odd Zumkeller numbers less than one million; they are exactly the odd abundant numbers that have even abundance. 

Today, my diurnal age is 25434, which is a Zumkellar number, and the next one is 25440. This is a difference of only six. From the earlier statistics, it can be seen that Zumkeller numbers occur with a frequency of about 22.9% for the first one million natural numbers.

Now OEIS A246198 defines half-Zumkeller numbers as numbers whose proper positive divisors can be partitioned into two disjoint sets whose sums are equal. The comment is made that all even half-Zumkeller numbers are in OEIS A083207, i.e. they are Zumkeller numbers. The first 47 terms coincide with A083207. 225 is the first number in the sequence that is not a Zumkeller number. The set of proper divisors of 225 is {1, 3, 5, 9, 15, 25, 45, 75} and this can be divided into two disjoint sets, {9, 75, 5} and {25, 3, 1, 45, 15} both totalling 89.

OEIS A005835 describes the pseudoperfect (or semiperfect) numbers as those in which some subset of the proper divisors of n sums to n. It's noted in the comments that deficient numbers cannot be pseudoperfect and that the first odd pseudoperfect number is 945. In the case of 945, the proper divisors are {1, 3, 5, 7, 9, 15, 21, 27, 35, 45, 63, 105, 135, 189, 315} and there are five subsets of these divisors that add to 945. Here is the list:

• {1, 9, 21, 27, 35, 45, 63, 105, 135, 189, 315}
• {3, 7, 21, 27, 35, 45, 63, 105, 135, 189, 315}
• {7, 9, 15, 27, 35, 45, 63, 105, 135, 189, 315}
• {1, 3, 5, 7, 15, 27, 35, 45, 63, 105, 135, 189, 315} 
• {1, 5, 7, 9, 15, 21, 35, 45, 63, 105, 135, 189, 315} 

While nearly all abundant numbers are pseudoperfect, some aren't. These numbers are termed weird and comprise OEIS A006037: weird numbers - abundant (A005101) but not pseudoperfect (A005835). From the comments to this sequence in the OEIS, we find:

Deléglise (1998) shows that abundant numbers have asymptotic density < 0.2480, resolving the question which he attributes to Henri Cohen of whether the abundant numbers have density greater or less than 1/4. The density of pseudoperfect numbers is the difference between the densities of abundant numbers (A005101) and weird numbers (A006037), since the remaining integers are perfect numbers (A000396), which have density 0. Using the first 22 primitive pseudoperfect numbers (A006036) and the fact that every multiple of a pseudoperfect number is pseudoperfect it can be shown that the density of pseudoperfect numbers is > 0.23790.

There are other interesting facts mentioned in the comments, including:

• The first weird number that has more than one decomposition of its divisors set into two subsets with equal sum (and thus is not a member of A083209) is 10430:

  1+5+7+10+14+35+298+10430 = 2+70+149+745+1043+1490+2086+5215

  2+70+298+10430 = 1+5+7+10+14+35+149+745+1043+1490+2086+5215.

• A weird number n multiplied with a prime \( p > \sigma(n) \) is again weird. Primitive weird numbers (A002975) are those which are not a multiple of a smaller term, i.e., don't have a weird proper divisor.
• No odd weird number exists below \(10^{21}\).

One thing to bear in mind about all the types of numbers mentioned above is that they are independent of the number system base used to depict the numbers. Other types of numbers are dependent on the number base. For example, a d-powerful number in base 10 is not necessarily a d-powerful number in another base.

Neil Sloane, the originator of the OEIS, has this to say about Reinhard Zumkellar, after whom the numbers are named:

I am deeply sorry to have to report that Reinhard Zumkeller passed away at the end of March 2016. He suffered from pancreatic cancer, which had already progressed to an advanced stage when it was diagnosed. He was a long-time contributor to the OEIS, and was later an editor and then a diligent and dedicated editor-in-chief. Between 2000 and 2016 he contributed over 23000 items to the OEIS. Reinhard was a great Haskell expert, and he was already ready to write a Haskell program and compute 10000 terms when I was studying a new sequence and wanted to see a graph. He will be greatly missed. Neil Sloane, July 3, 2016.