Anti-Perfect Numbers

The topic of anti-perfect numbers has everything to do with anti-divisors with parallels to perfect numbers and divisors. Let's recall that a perfect number is a number whose sum of proper divisors equals the number. The first perfect number is 6 and it is perfect because its proper divisors of 1, 2 and 3 add to 6. Similarly, an anti-perfect number is a number whose sum of anti-divisors equals the number. The first anti-perfect number is 5 because its anti-divisors of 2 and 3 add to 5.

Now I've written about anti-divisors before in two posts, one titled Anti-divisors on February 26th 2015 and another titled More on Anti-divisors on February 28th 2021. Today, my diurnal age is 27024 and this number has anti-divisors of 7, 32, 49, 96, 1103, 7721 and 18016. The sum of these anti-divisors is 27024 and so 27024 is an anti-perfect number. Like the perfect numbers, these anti-perfect numbers are few and far between. They constitute OEIS A073930:

A073930

Numbers that are equal to the sum of their anti-divisors.

Up to one million, the members of this sequence are 5, 8, 41, 56, 946, 5186, 6874, 8104, 17386, 27024, 84026, 167786, 2667584, 4921776. Thus 27024 is the last anti-perfect number that I'll encounter in my lifetime, hence my celebration of it. The perfect numbers are even sparcer: 6, 28, 496, 8128, 33550336, ... and so I've long ago also reached the last perfect number that I'll encounter in my lifetime.

Let's display the result again:$$ \underbrace{7+32+49+96+1103+7721+18016}_{\text{anti-divisors of }27024} =27024$$It can be noted that the largest anti-divisor, 18016, is exactly 2/3 of 27024 and the largest anti-divisor of any number is always either exactly 2/3 of that number or very close to it. Figure 1 shows a table of anti-perfect numbers and their anti-divisors.

Friendly versus Solitary Numbers

Today I turned 26190 days old and discovered that this number forms one half of a friendly pair of numbers. The other half is 8148. What do these two numbers have in common? Well, we find that:$$ \frac{\sigma_1(26190)}{26190}=\frac{70560}{26190}=\frac{784}{291} \text{ and } \frac{\sigma_1(8148)}{8148}=\frac{21952}{8148}=\frac{784}{291}$$So if the sum of the divisors of one number divided by that number is the same as the sum of the divisors of another divided by that other number, then the numbers are said to be friendly. Friendly numbers are not to be confused with amicable numbers where the numbers are related in such a way that the sum of the proper divisors of one is equal to the sum of the proper divisors of the other. The smallest pair of amicable numbers is 220 and 284. 

Getting back to friendly numbers, we find that friendly triples and higher-order tuples are also possible. Friendly triples include: 

• (2160, 5400, 13104)
• (9360, 21600, 23400)
• (4320, 4680, 26208)

Friendly quadruples include: 

• (6, 28, 496, 8128)
• (3612, 11610, 63984, 70434)
• (3948, 12690, 69936, 76986)

Friendly quintuples include:

• (84, 270, 1488, 1638, 24384)
• (30, 140, 2480, 6200, 40640)
• (420, 7440, 8190, 18600, 121920)

Numbers that have friends are called friendly numbers, and numbers that do not have friends are called solitary numbers.

This ratio of the sum-of-divisors of an integer \(n\) to the integer itself is termed its abundancy and is defined as: \( \displaystyle \frac{\sigma_1(n)}{n}\).

By this definition, two numbers are friendly is they have the same abundancy.  Two numbers with the same abundancy form a friendly pair; \(n\) numbers with the same abundancy form a friendly \(n\)-tuple. 

Abundancy may also be expressed as \( \sigma _{-1}(n)\) where \( \sigma _{k} \) denotes the sum of the \(k\)-th powers of the divisors of \(n\). When \(k\)=-1, we have the sum of the reciprocals of the divisors. The abundancy of a number \(n\) should not be confused with its abundance \( A(n) \equiv  \sigma_1(n)-2n \). Refer to WolframMathWorld.

From Wikipedia we learn that:

if the numbers \(n\) and \( \sigma(n) \) are coprime – meaning that the greatest common divisor of these numbers is 1, so that \( \sigma(n)/n \) is an irreducible fraction – then the number \(n\) is solitary. For a prime number \(p\), we have \( \sigma_1(p) = p + 1\), which is co-prime with \(p\).

Thus all primes and multiples of primes are solitary. Wikipedia continues:

No general method is known for determining whether a number is "friendly" or solitary. The smallest number whose classification is unknown is 10; it is conjectured to be solitary. If it is not, its smallest friend is at least \(10^{30}\). Small numbers with a relatively large smallest friend do exist: for instance, 24 is "friendly", with its smallest friend 91,963,648.

Mutually friendly numbers as we said earlier can form friendly \(n\)-tuples that might be considered families or clubs. It's an open question whether these families have an infinite number of members. For example, it is conjectured that there are infinitely many perfect numbers but only 51 are currently known. Each perfect number has an abundancy of 2 and thus currently the perfect numbers form a 51-tuple or a family with 51 members. 

Similarly multiply perfect numbers form friendly families but firstly let's define what is meant by a multiply perfect numbers:

For a given natural number \(k\), a number \(n\) is called \(k\)-perfect (or \(k\)-fold perfect) if and only if the sum of all positive divisors of \(n\) (the divisor function, \( \sigma(n) \), is equal to \(k \times n\); a number is thus perfect if and only if it is 2-perfect. A number that is \(k\)-perfect for a certain \(k\) is called a multiply perfect number. As of 2014, \(k\)-perfect numbers are known for each value of \(k\) up to 11. Source. Also see my blog post Multiperfect, Hyperfect and Superperfect Numbers from July 24th 2019.

The club of friendly numbers with abundancy equal to 9 has 2094 known members but these multiply perfect clubs or families are thought to be finite (unlike the perfect family that is conjectured to be infinite).

There are a number of OEIS sequences associated with friendly and solitary numbers. It was stated earlier that numbers that are coprime with their sum of divisors are solitary but this is sufficient and not necessary condition for solitariness. OEIS A095739 lists those numbers that are solitary and yet not coprime with their sum of divisors:

A095739

Numbers known to be solitary but not coprime to sigma.

The first of these numbers are 18, 45, 48, 52, 136, 148, 160, 162, 176, 192, 196, 208, 232, 244, 261, 272, 292, 296, 297, 304, 320, 352, 369, ...

26190, the number that began this post, is a member of OEIS A050973:

A050973

Larger member of friendly pairs ordered by smallest maximal element.

The initial member of this sequence are:

28, 140, 200, 224, 234, 270, 308, 364, 476, 496, 496, 532, 600, 644, 672, 700, 812, 819, 868, 936, 1036, 1148, 1170, 1204, 1316, 1400, 1484, 1488, 1488, 1540, 1638, 1638, 1638, 1652, 1708, 1800, 1820, 1876, 1988, 2016, 2044, 2200, 2212, 2324, ...

The smaller members of these pairs are given by OEIS A050972:

A050972

Smaller member of friendly pairs ordered by smallest maximal element.

The initial members of this sequence are:

6, 30, 80, 40, 12, 84, 66, 78, 102, 6, 28, 114, 240, 138, 120, 150, 174, 135, 186, 864, 222, 246, 60, 258, 282, 560, 318, 84, 270, 330, 84, 270, 1488, 354, 366, 720, 390, 402, 426, 360, 438, 880, 474, 498, 510, 440, 30, 140, 534, 132, 1040, 570, 582, 606, ...

From these sequences, we can form the various pairs e.g. 28 and 6, 140 and 30 etc. Notice the two numbers (819 and 135) marked in bold in the above sequences. This pair are an example of two odd numbers being friendly. There are also cases of even being friendly to odd, such as 42 and 544635 with abundancy 16/7.

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.