Amazing! Thanks so much for sharing this.

@M.Wind Good point. I think that’s a good one for a follow-up question.

@M.Wind Perhaps I’m mistaken, but GEdgar’s answer seems to indicate that there can’t be a longest-tailed distribution because any such distribution could be stretched into a longer one. If that’s the case, that would be a sufficient negative answer to the question.

@GEdgar Oh, sorry, I thought $p’$ was the derivative of $p$. Convert your comment to an answer?

@GEdgar Can you elaborate on what you’re proposing? I have a general background in mathematics but am not used to proving statements about the best functions of a particular type and could use some more detailed guidance. Thanks!

Oh, yes, sorry! I saw the if and only if but then missed the submonoid part. D’oh!

@GeoffreyTrang Let’s take $\mathbb{N} \setminus \{1, 9\}$. If I’m interpreting your comment correctly, this should be a submonoid of $\mathbb{N}$, but I don’t believe it is. Can you clarify?