Some Special Sums of Squares and Cubes

It's well known that some numbers can be written as the sum of two squares. The number 2 is the first such number because:$$2=1^2+1^2$$The first number that can be written as the sum of two distinct squares is 5 because:$$5=2^2+1^2$$However, let's consider the number 20 where we have:$$20 = 2^2 + 4^2$$What makes 20 special is that the divisors of 20 are 1, 2, 4, 5, 10 and 20. Two of the divisors, 2 and 4, form the base of the two squares that add together to total 20. This is the first such number with this property and, up to 40000, the other numbers are (permalink):

20, 80, 90, 180, 272, 320, 360, 468, 500, 650, 720, 810, 980, 1088, 1280, 1332, 1440, 1620, 1872, 2000, 2250, 2420, 2448, 2450, 2600, 2880, 2900, 3240, 3380, 3600, 3920, 4160, 4212, 4352, 4410, 4500, 5120, 5328, 5760, 5780, 5850, 6480, 6642, 6800, 7220, 7290, 7488, 7650, 8000, 8820, 9000, 9680, 9792, 9800, 10100, 10388, 10400, 10580, 10890, 11520, 11600, 11700, 11988, 12500, 12960, 13328, 13520, 14400, 14580, 14762, 15210, 15680, 16250, 16400, 16640, 16820, 16848, 17408, 17640, 18000, 19220, 20250, 20480, 20880, 21312, 21780, 22032, 22050, 22932, 23040, 23120, 23400, 24500, 25578, 25920, 26010, 26100, 26568, 27200, 27380, 27540, 28730, 28880, 29160, 29952, 30420, 30600, 31850, 32000, 32400, 32490, 32912, 33300, 33620, 35280, 36000, 36980, 37440, 37908, 38612, 38720, 39168, 39200, 39690

Looking more closely at these numbers it can be seen that some are multiples of smaller numbers. For example, consider the second number in the sequence: 80. We find that:$$ \begin{align} 80 &= 4^2+8^2\\ &=2^2(2^2+4^2) \\ &=4 \times 20 \end{align}$$Numbers like 20 are called primitive numbers and form OEIS A338485:

A338485

Primitive numbers that are the sum of the squares of two of their distinct divisors.

The members of this sequence up to 40000 are (permalink):

20, 90, 272, 468, 650, 1332, 2450, 2900, 3600, 4160, 6642, 7650, 10100, 10388, 14762, 16400, 20880, 25578, 27540, 28730, 38612

Let's look at the last member in the sequence above: 38612. We have:$$  38612 = 14^2+196^2$$The divisors of 38612 are 1, 2, 4, 7, 14, 28, 49, 98, 196, 197, 394, 788, 1379, 2758, 5516, 9653, 19306, 38612 and we can see that 14 and 196 are represented.

Figure 1 shows a list bases for the squares that form the primitive and non-primitive numbers from 27540 upwards.

Figure 1

20 is the first number than can be represented as the sum of squares of two of its divisors

I was naturally curious as to whether there were primitive numbers that are the sum of the cubes of two of their distinct divisors. There are indeed. The numbers, not necessarily primitive with this property, are up to 40000 (permalink):

72, 520, 576, 756, 1944, 4160, 4608, 6048, 7560, 9000, 14040, 15552, 15750, 19656, 19710, 20412, 24696, 32832, 33280, 36864

The primitive numbers up to 40000 are (permalink):

72, 520, 576, 756, 1944, 4160, 6048, 7560, 9000, 14040, 15552, 15750, 19656, 19710, 20412, 24696, 32832

The first number in the first list not to appear in the second list is 4608 which can be rendered as:$$ \begin{align} 4608 &= 8^3+16^3 \\  &=2^3 (4^3+8^3) \\ &=8 \times 576 \end{align}$$It can be seen then that 4608 is not a primitive number whereas 576 is. 576 has divisors of 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, 288 and 576 and 4 and 8 are included amongst these divisors.

The algorithm used to generate these lists is easily modified to accommodate fourth, fifth etc. powers if one is interested. I'll stop with the cubes in this post. Figure 2 shows what the bases of the cubes are for the primitive and non-primitive numbers.

Figure 2