Striking a Balance

Composite numbers have four or more divisors. For example, the number 6 has divisors of 1, 2, 3 and 6. The first three are deficient and the final divisor, the number itself, is perfect. Figure 1 shows the situation:

Figure 1: divisors of 6
permalink

The number 48 has divisors of 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48. The balance of deficient, perfect and abundant divisors is shown in Figure 2.

Figure 2: divisors of 48
permalink

The question can be asked as to what numbers have an equal balance of deficient and abundant divisors. It turns out that 144 is the first number to satisfy this criterion. The divisors of 144 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72 and 144. The balance of deficient, perfect and abundant divisors is shown in Figure 3.

Figure 3: divisors of 144
permalink

The numbers with an equal balance of deficient and abundant divisors constitute OEIS A335543 (permalink):

A335543

Numbers with an equal number of deficient and abundant divisors.

The initial members are 144, 324, 336, 756, 900, 1176, 1848, 2100, 2184, 2940, 3200, 3520, 4000, 4160, 4400, 5200, 5952, 10880, 11440, 12160, 12348, 12544, 13600, 14720, 15200, 16368, 17360, 18304, 18400, 18560, 19344, 19360, 19404, 22932, 23200, 27040, 28600, 29988, 33516, 40572, 47124.

It was only today that I became acquainted with this sequence because my diurnal age, 27040, happens to be a member. It's clear to see why I haven't come across the sequence before. The previous member is 23200 corresponding to a time when I wasn't keeping track of the numbers associated with my diurnal age. Figure 4 shows the breakdown for 27040 with divisors of 1, 2, 4, 5, 8, 10, 13, 16, 20, 26, 32, 40, 52, 65, 80, 104, 130, 160, 169, 208, 260, 338, 416, 520, 676, 845, 1040, 1352, 1690, 2080, 2704, 3380, 5408, 6760, 13520 and 27040.

Figure 4: divisors of 27040
permalink

As can be seen, these numbers are rare. There are only 39 of them in the range up to 40000. The OEIS entry has some interesting comments:

• This sequence is infinite. For example, \(3200 \times p \) is a term for all primes \(p \geq 257\). 

• The least odd term of this sequence is a(1273824) = 3010132125.

Checking out 3200 x 257 = 822400 we find that it does indeed have the required balance. See Figure 5:

Figure 5: divisors of 822400
permalink