Sums of Cubes and Squares of Sums

Today I turned 25321 days old, a prime number of days, and a prime with the property that the sum of the cubes of its digits equals the square of the sum of its digits. In other words:$$2^3+5^3+3^3+2^3+1^3 = (2+5+3+2+1)^2 = 169$$This prime is a member of OEIS A225567: Primes with nonzero digits such that sum of cubes of digits equal to square of sums. It's initial members are:$$1423, 2143, 2341, 4231, 12253, 21523, 22153, 22531, 23251, 25321, ...$$Serendipitously I then rediscovered the old and famous connection between the sum of the cubes of the first n natural numbers and the square of the sum of these some numbers, specifically:$$ \sum_1^n i^3=\big (\sum_1^n i \big)^2$$It's easy to see why the first four members of the sequence (1423, 2143, 2341 and 4231 are members) because these are simply instances of:$$1^3+2^3+3^3+4^3=(1+2+3+4)^2$$I also discovered a similar relationship involving the divisors \(d_i \) of any natural number with \(n\) divisors, namely that:$$ \sum_1^n (\sigma_0(d_i))^3=\big(\sum_1^n \sigma_0(d_i) \big) ^2$$The previous looks more difficult than it actually is and a simple example will assist. Let's consider the number 10. It has four divisors 1, 2, 5 and 10. Each of these divisors has 1, 2, 2 and 4 divisors respectively. We find that:$$1^3+2^3+2^3+4^3=(1+2+2+4)^2 = 81$$It's as simple as that and it applies to every natural number. Of course 1224 and permutations of these digits can be found in OEIS A227073: Positive numbers without the digit 0 such that sum of cubes of the digits equals the square of the sum of the digits. The initial members of this sequence are:$$ 1, 12, 21, 22, 123, 132, 213, 231, 312, 321, 333, 1224, ...$$

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 \( 432 \)