Deficient Numbers

Sometimes it's easy to forget the basics, such as what defines a deficient number. For example, today's number 25137 has the following entry in OEIS: deficient numbers n having a companion m > n such that sigma(n)/n = sigma(m)/m. The initial numbers in this sequence are shown below:

135, 3375, 1485, 2295, 2565, 3105, 3915, 4185, 4995, 5535, 5805, 6345, 25137, 7155, 7965, 8235, 9045, 9585, 9855, 10665, 11205, 12015, 13095, 13635, 13905, 14445, 14715, 43875, 15255, 16335, 17145, 17685, 18495, 18765, 57375, 20115, 20385, 21195, 64125

These numbers are listed in the order that their companions were found. All these numbers appear to have only one companion, which appear in A212609. The initial entries in this sequence are shown below with the 13th entry marked, namely 40131, because 25137 is the 13th entry in the previous set of numbers:

819, 6975, 9009, 13923, 15561, 18837, 23751, 25389, 30303, 33579, 35217, 38493, 40131, 43407, 48321, 49959, 54873, 58149, 59787, 64701, 67977, 72891, 79443, 82719, 84357, 87633, 89271, 90675, 92547, 99099, 104013, 107289, 112203, 113841, 118575, 122031

So the companion for 25137 is 40131 and checking we find that: $$ \frac {\sigma(25137)}{25137}=\frac{\sigma(40131)}{40131} \approx 1.81406 $$However, just to remind myself about the distinction between deficient, perfect and abundant numbers, I've included the following graphic:

By now the sigma function has begun to sink into my long term memory along with the Euler totient function or phi function as it's sometimes known.