4 ms·
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 parti
by isaacfrond 1y ago
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 https://arxiv.org/abs/2405.06451
- boothby 1y agoSorry, 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 )
- Sniffnoy 1y agoM_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 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!