  A primary pseudoperfect number (PPN) is an integer $K > 1$ such that the
reciprocals of $K$ and its prime factors sum to 1. PPNs arise in studying
perfectly weighted graphs and singularities of algebraic surfaces, and are
related to Sylvester's sequence, Giuga numbers, Zn\'am's problem, the
inheritance problem, and Curtiss's bound on solutions of a unit fraction
equation.
  Here we show $K \equiv 6 \pmod{6^2}$ if $6\mid K$, and uncover a remarkable
$7$-term arithmetic progression of residues modulo $6^2\cdot8$ in the sequence
of known PPNs. On that basis, we pose a conjecture which leads to a conditional
proof of the new record lower bound $k>10^{3.99\times10^{20}}$ on any
non-trivial solution to the Erd\H{o}s-Moser Diophantine equation $1^n + 2^n +
\dotsb + k^n = (k+1)^n$.
