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.