Relationship between primes and practical numbers

Solution 1:

The plot isn't actually linear, but it indeed looks like one. Here's why:

Let's introduce two functions, $\pi(x)$ and $p(x)$, that respectively count the number of prime and practical numbers less than $x$. A famous result in number theory, the prime number theorem, tells us that $$ \pi(x) \sim \frac{x}{\log(x)} \quad \text{for } x \to +\infty $$ which essentially means that for $x$ large enough $\pi(x)$ behaves almost exactly like $x/\log(x)$.¹ Remarkably, just in 2015 Weingartner showed that $$ p(x) \sim \frac{cx}{\log(x)} \quad \text{for } x \to +\infty $$ for some constant $c > 0$.

The sequence of partial sums you defined is then simply the sequence $\{s(n)\}_{n=1}^\infty$ where $s(x) := p(x) - \pi(x)$. It follows that $$ s(x) \sim (c-1) \frac{x}{\log(x)} \quad \text{for } x \to +\infty. $$ To conclude, here's what $\frac{x}{\log(x)}$ looks like on the interval $[0,10^4]$:

^{1. The almost is important: $\pi(x)$ may never be equal to $x/\log(x)$, but after a while the error will grow quite slower than $x/\log(x)$.}

Update: I should probably explicitly say something about your second graph, too, but first we need to give a precise meaning to "$\sim$". Given two functions $f,g \colon D \subseteq \Bbb{R} \to \Bbb{R}$ and $\rho \in \Bbb{R} \cup \{\pm\infty\}$, we say that $f \sim g$ for $x \to \rho$ if $$ \lim_{x \to \rho} \frac{f(x)}{g(x)} = 1. $$ Now, what can we say about the limit of the sequence depicted in your second graph? From the previous discussion we have $$ \begin{align} \lim_{n \to \infty} \frac{s(n)}{n} &= \lim_{n \to \infty} \frac{p(n)-\pi(n)}{n} \\ &= \lim_{n \to \infty} \frac{p(n)-\pi(n)}{n}\; \frac{\log(n)}{\log(n)} \\ &= \lim_{n \to \infty} \frac{p(n)-\pi(n)}{n/\log(n)}\; \frac{1}{\log(n)} \\ &= \left(\lim_{n \to \infty} \frac{p(n)-\pi(n)}{n/\log(n)}\right) \left(\lim_{n \to \infty}\frac{1}{\log(n)}\right) \\ &= (c-1) \lim_{n \to \infty}\frac{1}{\log(n)} \\ &= 0 \end{align} $$ So $s(n)/n$ does indeed converge to $0$, but I don't know how fast.