Minimal idempotency, partial idempotency, search heuristics and constructive algorithms for idempotent integers

Previous work established the set of square-free integers n with at least one factorization n=p ¯q ¯ for which p ¯ and q ¯ produce valid RSA keys, whether they are prime or composite. These integers are exactly those with the property λ(n=p ¯q ¯)∣(p ¯-1)(q ¯-1), where λ is the Carmichael totient function. We refer to these integers as idempotent, because ∀a∈Z n ,a k(p ¯-1)(q ¯-1)+1 ≡ n a for any positive integer k. This set includes the semiprimes and the Carmichael numbers, but is not limited to them. Numbers in this last category have not been previously analyzed in the literature.