on the weekend = American English at the weekend = British English We wouldn't say 'on a week.' 'in a week'= after one week 'Ill be back on Saturday - I'll see you in a week.' I hope to visit my sister during the week. We need to finish the job within a week.

"On" is reserved for individual days. In US (and Australian) English, we use "on the weekend", whereas in the UK "at the weekend" is used. For longer periods of time, such as weeks, months or years, you always use "in".