

A306746


A Goldbug number is an even number 2m for which there exists a subset of the prime nondivisors of 2m P={p1, p2, p3, ..., pk} where (2mp1)*(2mp2)*(2mp3)*...*(2mpk) has only elements of P as factors.


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. 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 equivalent to 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..6.
Craig J. Beisel, Maximal Sets of PNDs Satisfying Goldbug Property for First 6 Terms
Craig J. Beisel, Enumeration of all Goldbug subsets for the term 128.
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

If we examine the simplest term 2200 and the prime nondivisors 3 and 13 we can see that (2200  13)*(2200  3) = 4804839 = 3^7*13^3. Therefore 2200 is an order k=2 Goldbug since (3,13) is a subset of the prime nondivisors of 2200 such that the product (2200  3)*(2200  13) has only 3 and 13 as factors.
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. Therefore, 128 is a Goldbug number of order k=14.
It is interesting to note that Goldbug products are Solinas numbers which are not prime themselves but near other numbers which are hard to factor. For example, the Goldbug product (22003)(220013) = 75075*2^6 + 39 is not prime but is between the two Solinas primes 75075*2^6 + 37 and 75075*2^6 + 41 where the coefficients are coprime.


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

Shares 3 elements with A244408 (128, 1928, 2200). A057896
The Goldbug product 4804839 for 2200 is an element of A179705.
Sequence in context: A187706 A202961 A239540 * A269081 A200789 A250356
Adjacent sequences: A306743 A306744 A306745 * A306747 A306748 A306749


KEYWORD

nonn,more,hard


AUTHOR

Craig J. Beisel, Mar 07 2019


STATUS

approved



