Characteristic-Vector Congruences for Weak Primary Pseudoperfect and Giuga numbers

Abstract

Let $\mu\in\{-1,1\}$, and suppose that an integer $n>1$ satisfies \[a:=\frac{\mu}{n}+\sum_{p\mid n}\frac1p\in\mathbb Z.\] Writing $r=\omega(n)$ for the number of distinct prime divisors of $n$, we associate to $(n,a,\mu)$ an integral unimodular star lattice of determinant $\mu$ and signature $r+\mu$. We compute the norm of an explicit characteristic vector and apply van der Blij's congruence to prove \[r\equiv a-\mu\pmod8\] when $n$ is odd, and \[n\equiv4\mu(r-a)+4-2\mu\pmod{32}\] when $n$ is even. For $\mu=1$, these results give congruences for weak primary pseudoperfect numbers. In the primary pseudoperfect case $a=1$, the even congruence proves \[n\equiv4r-2\pmod{32},\] and consequently proves the conditional mod-$288$ progression conjectured by Sondow and MacMillan. The odd congruence implies that any odd primary pseudoperfect number has at least $16$ distinct prime factors. For $\mu=-1$, the theorem gives corresponding congruences for Giuga numbers. If $6\mid n$, then \[n\equiv102+36(a-r)\pmod{288}.\] In the index-one case, this accounts for the seven residue classes observed among the known Giuga numbers. It also shows that any such number satisfying $6\mid n$ and $r=9$ would satisfy $n\equiv102\pmod{288}$. We also prove that any odd Giuga number must have at least $18$ distinct prime factors.