A Goldbug number is an even number 2m for which there exists some subset of the prime non-divisors (PNDs) of 2m, 2 < p1 < p2 < p3 < ... < pk < m, such that (2m-p1)*(2m-p2)*(2m-p3)*...*(2m-pk) has only p1,p2,p3,...,pk as factors and pk > m/2. We do not need to consider the case where m is prime, since then (m,m) is itself a Goldbach pair. A Goldbug number is called "order-k" if the maximal subset satisfying the property is of size k.
These numbers arise from Goldbug's Algorithm which attempts to find a Goldbach pair for a particular even number by starting with a given PND p1 and successively adding the factors of the product (2m - p1)*...*(2m - pk) to the search until a pair is found. Goldbug numbers are those even numbers for which Goldbug's Algorithm is not guaranteed to find a Goldbach pair since it could reach a subset of the PNDs which does not contain new information about additional PNDs to add to the search.
Goldbug numbers are a special case of Basic Pipes as defined by Wu. It has been shown computationally a(6) > 5*10^8. See link.
Goldbug numbers serve as a link between Goldbach's conjecture and the Pillai conjecture since order-2 Goldbug numbers represent solutions to its generalized difference equation. For example, sequence
A057896 demonstrates there are no order 2 Goldbugs less than 10^24 since it would imply additional solutions to the equation a^x-a = b^y-b. In fact, theorem 3 from Scott[1993] implies that no additional order 2 Goldbugs exist at all.
