%I
%S 128,1718,1862,1928,6142
%N A Goldbug number is an even number 2m for which there exists a subset of the prime nondivisors, P={p1, p2, p3, ..., pk}, of 2m where (2mp1)*(2mp2)*(2mp3)*...*(2mpk) has only elements of P as factors and one of the pi is between n/2 and n for even n and between (n+1)/2 and n1 for odd n.
%C A Goldbug number is an even number 2m for which there exists some subset of the prime nondivisors (PNDs) of 2m, 2 < p1 < p2 < p3 < ... < pk < m, such that (2mp1)*(2mp2)*(2mp3)*...*(2mpk) has only p1,p2,p3,...,pk as factors and one of the pi is between n/2 and n for even n and between (n+1)/2 and n1 for odd n. We do not need to consider the case where n is prime, since then n itself is 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.
%C Goldbug numbers are a special case of Basic Pipes as defined by Wu. It has been shown computationally a(7) > 5*10^8. See link.
%C Sequence A057896 demonstrates there are no order 2 Goldbugs less than 10^24 since it would imply additional solutions to the equation a^xa = b^yb.
%H Craig J. Beisel, <a href="/A306746/a306746.txt">Maximal Sets of PNDs Satisfying Goldbug Property for First 5 Terms</a>
%H Craig J. Beisel, <a href="/A306746/a306746_1.txt">Enumeration of all Goldbug subsets for the term 128</a>.
%H Craig J. Beisel, <a href="/A306746/a306746_2.txt">Goldbug's Algorithm</a>.
%H Andrzej Bożek, <a href="https://arxiv.org/abs/1909.09900">Exceptional autonomous components of Goldbach factorization graphs</a>, arXiv:1909.09900 [math.NT], 2019.
%H Bert Dobbelaere, <a href="https://oeis.org/A306746/a306746.cpp.txt">C++ Program</a>.
%H Christian Goldbach, <a href="http://eulerarchive.maa.org/correspondence/letters/OO0765.pdf">Letter to L. Euler</a>, June 7, 1742.
%H Math Stack Exchange, <a href="https://math.stackexchange.com/questions/3026044/searchingforgoldbugnumbers">Searching for Goldbug Numbers</a>
%H Willie Wu, <a href="https://sites.google.com/site/basicpipetheory/">Pipe Theory</a>
%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%e Although 2200 and the prime nondivisors 3 and 13 might seem to satisfy the definition since (2200  13)*(2200  3) = 4804839 = 3^7*13^3, 2200 is not an order k=2 Goldbug since neither 3 or 13 is in the interval (n/2,n).
%e A higherorder example is the term 128, for which there exists a subset of the PNDs such that the corresponding product (128  3)*(128  5)*(128  7)*(128  11)*(128  13)*(128  17)*(128  23)*(128  29)*(128  37)*(128  41)*(128  43)*(128  47)*(128  53)*(128  59) = 8147166895749452778629296875 = (3^14)*(5^8)*(7^2)*(11^3)*(13^2)*17*(23^2)*29*37*41 and 37 and 41 are in the interval (32,64). Therefore, 128 is a Goldbug number of order k=14.
%o (PARI) isgbk(n,k) = {if (n % 2, return (0)); f=factor(n) [, 1]; vp = setminus(primes([3, n/2]), f~); forsubset([#vp,k], s, w=vecextract(vp, s); if(#w>1 && setminus(factor(x=prod(i=1, #s, nw[i]))[, 1]~, Set(w))==[], return(1)););return(0);} \\ tests if n is order k Goldbug;
%Y Cf. A244408, A057896.
%K nonn,more,hard
%O 1,1
%A _Craig J. Beisel_, Mar 07 2019
