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.

432 Hz versus 440 Hz

I've been aware for a while about the the controversy surrounding the standard A note and whether it should be set to \(440 \text{Hz} \) (as it now is) or changed to \( 432 \text{Hz}. \) I'm trying in this post to look at the mathematical properties of \( 432 \).

• \( 432^2 = 186624 \) is close to the speed of light as measured in miles per second. Wolfram Alpha gives a figure of \( 186282 \) miles per second for the speed of light in a vacuum which is \( 99.82 \text{%} \) of \( 432^2 \).
  
  
• It also turns out that the area of an equilateral triangle whose numerical area is equal to its perimeter is given by \(12 \sqrt{3} = \sqrt{432} \).
  
  
• \( 432 \) sits between the twin primes \( 431 \) and \( 433 \)
  
  
• The factors of \( 432 \) are \( 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 48, 54, 72, 108, 144, 216 \text{ and } 432 \). The sum of these divisors is \(1240 \).
  
  
• \(432 \) is a 3-smooth number, one that is of the form \( 2^i*3^j \text{ where }i,j>=0 \) or to put it less mathematically it is a number that can be written as a power of two times a power of three, specifically \( 2^4×3^3 \). Such numbers have been called harmonic numbers. Here are the harmonic numbers up to \( 1000 \):
  
  

• \( 432 \) is the sum of four consecutive primes: \(103+107+109+113 = 432\)
  
  
• \( 432 \) is the sum of two positive cubes: \( 6^3+6^3=432 \)
  
  
• OEIS lists \( 2944 \) entries for the number \( 432 \)