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