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> 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

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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

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