Equivalently,N is a primary pseudoperfect number if it satisfies
Except for the primary pseudoperfect numberN = 2, this expression gives a representation forN as the sum of distinct divisors ofN. Therefore, each primary pseudoperfect numberN (exceptN = 2) is alsopseudoperfect.
The eight known primary pseudoperfect numbers are
2,6,42, 1806, 47058, 2214502422, 52495396602, 8490421583559688410706771261086 (sequenceA054377 in theOEIS).
The first four of these numbers are one less than the corresponding numbers inSylvester's sequence, but then the twosequences diverge.
It is unknown whether there are infinitely many primary pseudoperfect numbers, or whether there are anyodd primary pseudoperfect numbers.
The prime factors of primary pseudoperfect numbers sometimes may provide solutions toZnám's problem, in which all elements of the solution set are prime. For instance, the prime factors of the primary pseudoperfect number 47058 form the solution set {2,3,11,23,31} to Znám's problem. However, the smaller primary pseudoperfect numbers 2, 6, 42, and 1806 do not correspond to solutions to Znám's problem in this way, as their sets of prime factors violate the requirement that no number in the set can equal one plus the product of the other numbers. Anne (1998) observes that there is exactly one solution set of this type that hask primes in it, for eachk ≤ 8, andconjectures that the same is true for largerk.
If a primary pseudoperfect numberN is one less than a prime number, thenN × (N + 1) is also primary pseudoperfect. For instance, 47058 is primary pseudoperfect, and 47059 is prime, so 47058 × 47059 = 2214502422 is also primary pseudoperfect.
Primary pseudoperfect numbers were first investigated and named by Butske, Jaje, and Mayernik (2000). Using computational search techniques, they proved the remarkable result that for each positiveintegerr up to 8, there exists exactly one primary pseudoperfect number with preciselyr (distinct) prime factors, namely, therth known primary pseudoperfect number. Those with 2 ≤r ≤ 8, when reducedmodulo 288, form thearithmetic progression 6, 42, 78, 114, 150, 186, 222, as was observed by Sondow and MacMillan (2017).
