Number of terms of sequence under $n$ with no arithmetic bounds.

Question

I mentioned to a friend recently the result that for any $n > 1$, there is an $N$ beyond which there is always a prime between $m$ and $nm$ (for $m>N$, if it wasn't clear).

They proclaimed "This is very intuitive, since by choosing a large enough $m$, the gap between $m$ and $nm$ will be also large and so with an infinite number of primes, it only makes sense that this would be the case." This heuristic makes sense for things like primes but I, not thinking this was intuitive in general, wondered if there were infinite sequences for which no such bounds exist...

Can you give examples of sequences where either:

a) $\exists n$ for which there is no $N$ satisfying the above criteria or preferably

b) $\nexists n$ for which there is an $N$ satisfying the above criteria?

I was thinking that at least a) could be shown for some doubly exponential sequences via generating function magic, but was not sure.

Answer 1

How about the sequence of factorials? For any $n>1$, you reach a point where you have to multiply by a number greater than $n$ to get to the next factorial.

The condition you're looking at is, that after taking logarithms, terms occur within each interval of a given size. (This can be seen because $\log(mn)=\log(m)+\log(n)$, so every multiplication by $n$ just amounts to adding $\log(n)$.) However, even after taking logarithms, the distances between factorials grows without bound.

Answer 2

It depends on how large a gap you consider to be intuitive.

Wikipedia's article on prime gaps (https://en.wikipedia.org/wiki/Prime_gap) has a number of interesting results and conjectures including this:

"In 1931, E. Westzynthius proved that maximal prime gaps grow more than logarithmically. That is,[9]

${\displaystyle \limsup _{n\to \infty }{\frac {g_{n}}{\log p_{n}}}=\infty .} $"