Characteristic-Vector Congruences for Weak Primary Pseudoperfect and Giuga numbers

Let μ∈\-1,1\, and suppose that an integer $n>1$ satisfies \[a:=μ/n+∑p n1p∈ Z.\] Writing r=ω(n) for the number of distinct prime divisors of n, we associate to (n,a,μ) an integral unimodular star lattice of determinant μ and signature r+μ. We compute the norm of an explicit characteristic vector and apply van der Blij's congruence to prove \[r≡ a-μ8\] when n is odd, and \[n≡4μ(r-a)+4-2μ{32}\] when n is even. For μ=1, these results give congruences for weak primary pseudoperfect numbers. In the primary pseudoperfect case $a=1$, the even congruence proves \[n≡4r-2{32},\] 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 μ=-1, the theorem gives corresponding congruences for Giuga numbers. If 6 n, then \[n≡102+36(a-r){288}.\] 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 n and $r=9$ would satisfy n≡102288. We also prove that any odd Giuga number must have at least $18$ distinct prime factors.