In the recent paper arXiv:2405.06451 the authors provide infinitely many characterizations of the primes using MacMahon partition functions: for $a>0$ the functions $M_a(n):=\sum\limits_{0<s_1<s_2<\dots<s_a\atop n=m_1s_1+m_2s_2+\dots +m_as_a}m_1m_2\dots m_a$
The simplest is : $n$ is prime iff $(n^2-3n+2)M_1(n)-8M_2(n)=0$
I'm not really able to compute those functions. Could someone please write out the above characterization in full for the first few $n=2,3,4,5,6$ ?
Added:
For $n=3$, which is prime, what would make sense is $M_1(3)=1+3=4$ since $3=3×1=1×3$ and $M_2(3)=1×1=1$ since $3=1×1+1×2$, and indeed we have the equality $(3^2-3×3+2)×4-8×1=0$.
For $n=4$, which is composite, since $4=4×1=2×2=1×4$ we should have $M_1(4)=1+2+4=7$ and since $4=1×1+1×3=2×1+1×2$ we should have $M_2(4)=3+2=5$ and indeed $(4^2-3×4+2)×7-8×5=6×7-8×5=2$ is non zero positive.
Could someone please confirm?