Unitary Divisors

My diurnal age today is 25245 and the OEIS A127666 mentions this number as belonging to the sequence of odd infinitary abundant numbers. Unfortunately, I had no idea what was meant by the term infinitary but I was determined to find out. This led me into deeper waters very quickly and I realised that it might be best to start in the paddle pool first by investigating the term unitary.

To quote from Wikipedia:

A natural number 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 60/10 have a common factor other than 1, namely 2. 1 is a unitary divisor of every natural number. 

Equivalently, a given divisor a of b is a unitary divisor if and only if every prime factor of a has the same multiplicity in a as it has in b. 

The sum of unitary divisors function is denoted by the lowercase Greek letter sigma thus: \( \sigma  ^*(n) \). The sum of the \(k \, th \) powers of the unitary divisors is denoted by \( \sigma_k ^*(n) \):$$ \sigma_k^*(n) = \sum_{d\mid n \atop \gcd(d,n/d)=1} d^k $$ If the proper unitary divisors of a given number add up to that number, then that number is called a unitary perfect number.

Now 60 turns out to be a unitary perfect number, one of very few in fact that are known. The list runs:

6, 60, 90, 87360, 146361946186458562560000 

"Some perfect numbers are not unitary perfect numbers, and some unitary perfect numbers are not regular perfect numbers ... It is not known whether or not there are infinitely many unitary perfect numbers, or indeed whether there are any further examples beyond the five already known" from Wikipedia.

The program code I used to determine the unitary divisors of a number in SageMath is (permalink):

#FIND UNITARY DIVISORS AND TEST FOR UNITARY PERFECT NUMBERS

number = 87360

total = 0

D = divisors(number)

print("Unitary Divisors are:")

for i in range(0, len(D)):

    if gcd(D[i], number/D[i]) == 1:

        print(D[i], end=" ")

        total = total+D[i]

print()

print("Sum of unitary divisors less",number,"is",total-number)

if total-number == number:

    print("Therefore",number,"is a unitary perfect number")

else:

    print("Therefore",number,"is not a unitary perfect number")

Unitary Divisors are:

1 3 5 7 13 15 21 35 39 64 65 91 105 192 195 273 320 448 455 832 960 1344 1365 2240 2496 4160 5824 6720 12480 17472 29120 87360 

Sum of unitary divisors less 87360 is 87360

Therefore 87360 is a unitary perfect number

I started off this post by investigating the term unitary and ended up focusing on unitary divisors rather than unitary numbers. Let's address that deficiency now. If the sum of the unitary divisors of a number is greater than the number, the number can be described as a unitary abundant number. Such numbers can be odd or even, with the former being much rarer. If the sum of the unitary divisors is less than the number, the number can be described as a unitary deficient number. As we've seen, if the sum of the unitary divisors is equal to the number, the number can be described as a unitary perfect number.

Of course, I've not examined in this post what are meant by infinitary divisors. That will have to wait for a future post.