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Comment by tgb | Hacker News Reader

Or primorials, which generally have lots of divisors compared to their size. Primorials go 123251723111... Basically just include the 'new' prime factors in your factorial. But even these aren't necessarily "the best" for getting lots of divisors for their size since, well, for example you multiple by 11 maybe you should multiply by a another 2 instead - does the reduced size more than compensate from the reduced divisors?

https://en.wikipedia.org/wiki/Primorial

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The HN markdown is interpreting your asterisks a signals to start/stop italics. You need to escape them with a \ first.

HN most assuredly does nothing like markdown for comment rendering.

It doesn't use Markdown(TM)* , but I have often seen "markdown" used as an informal term for the stripped down text formatting available in comments on websites. It's probably technically incorrect, but I'm not aware of a more accurate but similarly short name for it.

* I'm not actually sure Markdown is trademarked

Also, it turns out that you can't even escape asterisks with a \, you need to leave whitespace after them. So my original comment was doubly wrong.

> But even these aren't necessarily "the best" for getting lots of divisors for their size since, well, for example you multiple by 11 maybe you should multiply by a another 2 instead

The concept you may be reaching for here is Highly Composite Numbers. https://en.wikipedia.org/wiki/Highly_composite_number

While highly composite numbers relate to the number of divisors, an even closer fit to the Robin inequality are the somewhat similar superabundant and colossally abundant numbers, which relate to the sum of divisors instead.

https://en.wikipedia.org/wiki/Superabundant_number

In fact if counterexamples to the inequality exist, the smallest such counterexample must be a superabundant number.

This is a nice, accessible paper about searching for counterexamples to the inequality by generating superabundant and colossally abundant numbers: https://projecteuclid.org/euclid.em/1175789744

It might be not-super-difficult to prove that primorials larger than 5040 always do satisfy the inequality, due to the fairly simple structure of primorials. I was never great at analysis, so I don't see how to do it offhand, but I suspect it is significantly easier than the full RH.

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