I need help identifying this formula please A few months ago, I became interested in time series analysis and started digging through statistics formulas. I found a formula that I ended up playing around with, but I've misplaced my notes and cannot remember what it's called or what its purpose is. I'm not a mathematician (I'm a programmer), so I'll write it out here using some pseudo code.
Let's say we have 2 numbers, A and B:
(A+B)/sqrt((A^2+B^2)*2)
... or let's say we have 3 numbers, A, B, and C
(A+B+C)/sqrt((A^2+B^2+C^2)*3)
... etc
You might also write it like this:
sum(A,B,C) / sqrt( sumsq(A,B,C)*count(A,B,C) )
 A: Without knowing for sure what the formula that you reference is, if I assume that $A,B,C,\ldots\in\mathbb{R}_{\geq0}$, then it appears the formula is some measure of the similarity of the elements in a set of numbers. The formula you gave is
$$\frac{\sum_{i=1}^{n}x_{i}}{\sqrt{n\sum_{i=1}^{n}x_{i}^{2}}}$$
where we assume that there are $n$ numbers provided. We can see that for a set of $n$ equal numbers we get
$$\frac{nx}{\sqrt{n\cdot nx^{2}}}=\frac{nx}{nx}=1$$
As you can see, for any set of $n$ equal numbers, we get a measure of 1. This is the upper bound on the similarity measure, for any set of $n$ numbers. Similarly, if we examine the trivial case where, for a set of $n$ numbers, $(n-1)$ of them are zero and the other is non-zero we get
$$\frac{a}{\sqrt{n\cdot a^{2}}}=\frac{1}{\sqrt{n}}$$
which approaches 0 as $n\rightarrow\infty$. This means that for a given $n$, the lower bound on the similarity measure is given by the above. A set of very different numbers will approach this bound quickly.
The measure is not defined for the set $\mathbf{0}_{n}$.
For completeness, let's look at a few examples of sets of numbers $\mathbf{X}_{n}$ and the corresponding measure $\alpha$:
$$\begin{align}
\mathbf{X}_{2}&=(a,b)=(4,4)\Rightarrow\alpha=\frac{8}{\sqrt{64}}=1\\
\mathbf{X}_{2}&=(a,b)=(2,10)\Rightarrow\alpha=\frac{12}{\sqrt{208}}\approx 0.8321\\
\mathbf{X}_{5}&=(a,b,c,d,e)=(2,4,6,8,10)\Rightarrow\alpha=\frac{30}{\sqrt{1100}}\approx 0.9045\\
\mathbf{X}_{5}&=(a,b,c,d,e)=(2,0,0,0,0)\Rightarrow\alpha=\frac{2}{\sqrt{20}}=\frac{1}{\sqrt{5}}\approx 0.4472\\
\mathbf{X}_{5}&=(a,b,c,d,e)=(1,10,100,1000,10000)\Rightarrow\alpha=\frac{11111}{\sqrt{505050505}}\approx 0.4944>1/\sqrt{5}
\end{align}$$
Furthermore, the measure is scale invariant i.e. 
$$\begin{align}
\alpha_{\mathbf{X}_{n}}&=\alpha_{y\cdot\mathbf{X}_{n}}
\end{align}$$
for some $y>0$.
However, the measure is heavily influenced by shifts in the vector i.e.
$$\begin{align}
\alpha_{\mathbf{X}_{n}}&<\alpha_{y+\mathbf{X}_{n}}
\end{align}$$
for some $y>0$. As the difference between the values of the elements of the set becomes small relative to the absolute size of the values of the elements, the measure approaches 1.
