Gaps between Deficient Numbers

An alternative title for this post could have been Runs of Abundant Numbers because the two topics are complementary. Today I turned 26143 days old and one of this number's properties is that it's a member of OEIS A317049.

A317049

Numbers \(k\) such that both \(k\) and \(k\) + 3 are consecutive deficient numbers.

At first, this didn't seem all that significant a property, until one looks at the sequence and realises that this is a relatively rare occurrence. Below are the members less than 100,000:

5774, 5983, 7423, 11023, 21734, 21943, 26143, 27403, 39374, 43063, 49663, 56923, 58694, 61423, 69614, 70783, 76543, 77174, 79694, 81079, 81674, 82003, 84523, 84643, 89774, 91663, 98174, ...

Figure 1

As Figure 1 shows, the usual pattern is a run of deficient numbers punctuated by an abundant number. This is because approximately three out of every four numbers will be deficient. The following SageMath algorithm (permalink) will generate the above sequence of numbers:

L=[]

gap=3

for n in [1..100000]:

    N=[]

    for i in [0..gap]:

        difference=(n+i)-(sigma(n+i)-(n+i))

        N.append(difference)

    if N[0]>0 and N[gap]>0:

        OK=1

        if gap>1:

            for i in [1..(gap-1)]:

                if N[i]>0:

                    OK=0

        if OK==1:

            L.append(n)

print(L)

The obvious question then is where do runs of three abundant numbers occur or where do we find numbers \(k\) such that both \(k\) and \(k\) + 4 are consecutive deficient numbers. Well this occurs between between 171078829 and 171078833, where these two are consecutive deficient numbers. In other words, the consecutive abundant numbers are 171078830, 171078831 and 171078832.

The starting term of the smallest consecutive 4-tuple of abundant numbers is at most:

141363708067871564084949719820472453374

and so 141363708067871564084949719820472453373 to 141363708067871564084949719820472453378 is probably the smallest \(k\) to \(k\)+5 case.

See OEIS A094268 for more information. Thus it will be another 1260 days before there is another run of two abundant numbers. Note that most abundant numbers are even, so if two abundant numbers are to be adjacent then one of them must be odd which is rare. Returning to the deficient 26143 (the number that prompted this post), it can be noted that 26144 is even and abundant while 26145 is odd and abundant. Furthermore, 26145 is an odd primitive abundant number, meaning that none of its proper divisors is abundant. Such numbers form OEIS sequence A006038:

A006038

Odd primitive abundant numbers.

The sequence, up to 26145, runs:

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

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.

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.