Primary Pseudoperfect Numbers, Arithmetic Progressions, and the Erdős-Moser Equation

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, Znam's problem, and Curtiss's bound on solutions of a unit fraction equation. In this paper, we show that K is congruent to 6 modulo 36 if 6 divides K, and uncover a remarkable 7-term arithmetic progression of residues modulo 288 in the sequence of known PPNs. On that basis, we pose a conjecture which leads to a conditional proof of a new record lower bound on any nontrivial solution to the Erdos-Moser Diophantine equation.