If the company has a professional investigator run a background check then every job where taxes have been paid (social security, etc.) will show. Schools in another thing, some private schools will not show, but if you have obtained a student loan that will show.
Some companies do not obtain an extensive background check that shows everything. That depends mainly on the job they are hiring for.
For the job you had for a week I would think that if the company ever asked you why you did not list it, all you say is that since it was such a short time you felt it was insignificant. Good luck.