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Comment by Sniffnoy | Hacker News Reader

I'm a little confused at the significance here. Before I read the definition of the M_a, this seemed crazy, but on actually reading it, M_1 is just the sum-of-divisors function (usually denoted sigma).

So, n is prime iff M_1(n)=n+1. That's much simpler than the first equation listed there!

Indeed, looking things up, it seems that in general the functions M_a can be written as a linear combination (note: with polynomial coefficients, not constant) of the sigma_k (sigma_k is the sum of the k'th power of the divisors). So this result becomes a lot less surprising once you know that...

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The M functions are the MacMahon’s partition functions (see the paper [1]). They were not known to relate to the sum of divisors. The M_a function counts partitions in a parts but weighing multiplicities in the partion.

[1]: https://arxiv.org/abs/2405.06451

Sorry, but M_1 is simply the sum of divisors, and I don't think that was ever a mystery. Specializing the notation from the paper for M_a, to a=1, and writing pythonic with finite bounds for clarity...

  M_1(n) = sum(
    m
    for m in range(1, n+1)
    for s in range(1, n+1)
    if m*s = n
  )

M_1 is obviously just sigma. That's straight from the definition, you can't tell me that wasn't known.

As for the higher ones, I'm having trouble finding a proper citation saying that this was known earlier, but this math.stackexchange answer asserts that MacMahon himself worked some of this out: https://math.stackexchange.com/a/4922496/2884 No proper citation though, annoying.

When you say "this wasn't known", on what basis is that? It's very hard to be sure that something wasn't known unless you're an expert on that particular thing!

I agree that the observation "M_1(n) = n+1 iff n is prime" is elementary. It certainly motivates some intuition behind the investigation in this paper, but I'd loathe to call it obvious.

Note that the paper studies equations with polynomial coefficients on McMahon series. That is, the n+1 in our trivial observation is "stray" in a sense.

For an at-a-glance indication of nontriviality, look no further than the conjecture associated with Theorem 1.2 -- that there are exactly five equations of this sort which are prime indicators. That seems spooky, to me; I can't help but wonder what structure underlies such a small number of relations.

Can you elaborate? How does this result become less surprising if you know that? Personally I would not have guessed that there are infinitely many characterisations of P involving sums-of-powers-of-divisors either.

I mean, if you can do something a simple way, it's not that surprising that you can also do it a complicated way, I'd say.

It's not surprising you can do exactly the same thing in more complex ways... but we're talking about infinitely many independent characterisations of P here.

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