I've searched and cannot find this pattern anywhere concerning integers and their factors Over the last few months, I've been studying a pattern that I stumbled on concerning integers and their factors. First, I noticed that the number of factors a number has, follows an extremely regular pattern based on prime numbers. Meaning that starting with any prime and following multiples of that prime, the number of factors in those multiples will be the same. There are exceptions, but they are predictable. Each column below represents a prime number base and the number of factors for the prime multiple. It contains the primes from 2-199. I haven't seen this anywhere after a bit of searching. 


Your 'less interesting' pattern has a very simple explanation.
The number of divisors $d(n)$ for a number $n = p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is equal to $(e_1+1)(e_2+2)\cdots(e_k+1)$. This includes the $1$ and $n$ divisors, which is standard mathematical convention.
That means that any multiple $pk$ of a prime number $p$ will have $d(p)d(k) = 2d(k)$ divisors unless $p$ divides $k$, which is fairly rare for big $p$.
The sum of divisors of $n$ is $\displaystyle \sigma(n) = \prod_{i=1}^k \frac{p_i^{e_i + 1} - 1}{p_i - 1}$. Here as well we find that if $k$ and $p$ are coprime then the sum of divisors multiple $\sigma(kp) = \sigma(k)\sigma(p) = \sigma(k)(p+1)$. That's why for every row, when $\sigma(k)$ is big, the entire row is bright.
