Primary pseudoperfect number

For instance, the prime factors of the primary pseudoperfect number 47058 form the solution set {2,3,11,23,31} to Znám's problem.

Anne (1998) observes that there is exactly one solution set of this type that has k primes in it, for each k ≤ 8, and conjectures that the same is true for larger k. If a primary pseudoperfect number N is one less than a prime number, then N × (N + 1) 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 positive integer r up to 8, there exists exactly one primary pseudoperfect number with precisely r (distinct) prime factors, namely, the rth known primary pseudoperfect number.

Those with 2 ≤ r ≤ 8, when reduced modulo 288, form the arithmetic progression 6, 42, 78, 114, 150, 186, 222, as was observed by Sondow and MacMillan (2017).