Root-Mean-Square And Other Means

Today I turned 26095 days old and this number happens to be a so-called RMS number where RMS stands for Root-Mean-Square. Such numbers are defined by OEIS A140480 as numbers \(n\) such that root mean square of divisors of \(n\) is an integer. Now the root mean square of divisors is defined by MathWorld as:

For a set of \(n\) numbers or values of a discrete distribution \(x_i, ..., x_n\), the root-mean-square (abbreviated "RMS" and sometimes called the quadratic mean), is the square root of mean of the values \(x_i^2\), namely:$$x_{RMS}=\sqrt{\frac{x_1^2+x_2^2+...+x_n^2}{n}}=\sqrt{\dfrac{\sum \limits_{i=1}^n x_i^2}{n}}$$For a variate \(\chi\) from a continuous distribution \(P(x)\), we have:$$x_{RMS}=\sqrt{ \frac{\int[P(x)]^2 dx}{\int P(x) \,dx}}$$where the integrals are taken over the domain of the distribution. Similarly, for a function \(f(t)\) periodic over the interval \([T_1,T_2]\), the root-mean-square is defined as:$$f_{RMS}=\sqrt{\frac{1}{T_2-T_1}\int_{T_1}^{T_2} [f(t)]^2 dt}$$

The sequence to which 26095 belongs runs 1, 7, 41, 239, 287, 1673, 3055, 6665, 9545, 9799, 9855, 21385, 26095, ... with 7, 41 and 239 being prime and the next prime being 9369319. Such primes are known as NSW primes, after Newman, Shanks, and Williams (the authors of a paper on the subject back in 1981). If we designate \(n\) to be such a prime number, then \(n\) has 2 divisors \([1, n]\) and we have to solve Pell's equation \(n^2 = 2*C^2 - 1\) where \(C\) is a positive integer. The solution is a prime \(n\) of the form \(u_i = 6u_{i-1} - u_{i-2} \), where \(i \geq 2, u_0=1, u_1=7\). These primes are listed in OEIS A088165.

There are of course many other types of means including arithmetic, geometric, harmonic, Pythagorean, power, Heronian, Identric, population, Chisini, Stolarsky, Lehmer, weighted and so on. The arithmetic mean is certainly the best known and most widely used but the root-mean-square has many applications in scientific circles. I've encountered the root-mean-square before in the context of the root-mean-square-error (see Figure 1).

Figure 1

A regression line is a line drawn such that the RMSE is minimised (see Figure 2).

Figure 2

The root-mean-square is a particular instance of a more generalised power mean defined as:$$M_p(a_1, a_2, ..., a_n) \equiv \bigg( \frac{1}{n} \sum_{k=1}^n a_k^{\,p} \bigg)^{1/p} $$where the parameter \(p\) is an affinely extended real number and all \(a_k \geq 0\). A power mean is also known as a generalized mean, Hölder mean, or mean of degree (or order or power) \(p\). The case of \(p=1\) is the arithmetic mean and the case of \(p=2\) is the root-mean-square. 

Figure 3 shows a summary of a few particular values of \(p\) that yield special cases with their own names (source):

Figure 3

The three "classic" means \(A\) (the arithmetic mean), \(G\) (the geometric mean), and \(H\) (the harmonic mean) are sometimes known as the Pythagorean means. Figure 4 shows how these means on two elements \(a\) and \(b\) could be constructed geometrically, and also demonstrates that \(H \leq G \leq A\).

Figure 4: source