Does the sum of the reciprocal highly composite numbers converge or diverge?

Question

Does the sum $1/1 + 1/2 + 1/4 + 1/6 +\dots$ converge or diverge, given it's the sum of the reciprocals of the Highly composite numbers?

List of highly composite numbers: https://oeis.org/A002182 . . . I've tried to do some approximations, but I'm unsure, considering there is a relatively small amount of highly composites even up to millions of digits.

I'm assuming that its probably pretty hard to conclusively show this series converges or diverges - and am mostly looking for anyone's thoughts on if it probably converges or diverges.

Answer 1

Let $a_n$ denote the $n$-th highly composite number (A002182). You observed that “... there is a relatively small amount of highly composites even up to millions of digits.” That is because the sequence $(a_n)$ grows “fast” and that makes it likely that $\sum 1/a_n$ is convergent.

In order to make this rigorous one can try to find a lower bound for $a_n$. There are surely more precise estimates of the asymptotics, but the following can be obtained with elementary means and is good enough to prove the convergence:

$a_n$ has at least $n$ divisors. On the other hand, the “number of divisors function” $d(n)$ satisfies $d(n) \le 2 \sqrt n$, see for example Upper limit for the Divisor function.

Therefore $$ n \le d(a_n) \le 2 \sqrt{a_n} \implies a_n \ge \frac 14 n^2 $$ so that $\sum_{n=1}^\infty \frac{1}{a_n}$ is convergent.

See also A352418 for the decimal expansion of the sum of the reciprocals of the highly composite numbers: $$ \sum_{n=1}^\infty \frac{1}{a_n} = 2.13287282352581201034387841183051877907288806895933... $$

Answer 2

Given $d \ge 1$, find $a(d)$ which is the smallest number with $d$ divisors. For every $d$, $a(d)$ is the only number that might be a highly composite number with $d$ divisors (for example $a(3) = 8$ is not highly composite).

You can calculate that $a(d)$ grows quite fast with $d$, so the sum of $1 / a(d)$ is bounded, and the sum of the reciprocals of highly composite numbers is less.