

A306746


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.


2




OFFSET

1,1


COMMENTS

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.
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.
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.


LINKS

Table of n, a(n) for n=1..5.
Craig J. Beisel, Maximal Sets of PNDs Satisfying Goldbug Property for First 5 Terms
Craig J. Beisel, Enumeration of all Goldbug subsets for the term 128.
Craig J. Beisel, Goldbug's Algorithm.
Andrzej Bożek, Exceptional autonomous components of Goldbach factorization graphs, arXiv:1909.09900 [math.NT], 2019.
Bert Dobbelaere, C++ Program.
Christian Goldbach, Letter to L. Euler, June 7, 1742.
Math Stack Exchange, Searching for Goldbug Numbers
Willie Wu, Pipe Theory
Index entries for sequences related to Goldbach conjecture


EXAMPLE

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).
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.


PROG

(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;


CROSSREFS

Cf. A244408, A057896.
Sequence in context: A202961 A239540 A327772 * A332545 A269081 A200789
Adjacent sequences: A306743 A306744 A306745 * A306747 A306748 A306749


KEYWORD

nonn,more,hard


AUTHOR

Craig J. Beisel, Mar 07 2019


STATUS

approved



