primary pseudoperfect number

Given an integer $n$ with $\omega (n)$ distinct prime factors $p_i$ (where $\omega$ is number of distinct prime factors function), if the equality

$$\frac{1}{n}+\sum _{i=1}^{\omega (n)}\frac{1}{p_i}=1$$

holds true, then $n$ is a primary pseudoperfect number. Equivalently,

$$1+\sum _{i=1}^{\omega (n)}\frac{n}{p_i}=n$$

if $n$ is a primary pseudoperfect number.

The first few primary pseudoperfect numbers are 2, 6, 42, 1806, 47058, 2214502422, 52495396602, 8490421583559688410706771261086, the first four of these being each one less than the first four terms of Sylvester’s sequence; these are listed in A054377 of Sloane’s OEIS. Presently it’s not known whether there are any odd primary pseudoperfect numbers.

Title	primary pseudoperfect number
Canonical name	PrimaryPseudoperfectNumber
Date of creation	2013-03-22 16:17:40
Last modified on	2013-03-22 16:17:40
Owner	CompositeFan (12809)
Last modified by	CompositeFan (12809)
Numerical id	6
Author	CompositeFan (12809)
Entry type	Definition
Classification	msc 11D85
Related topic	GiugaNumber