Simple day-of-week program
Your DayOnWeek object is mutable, allowing you to set the .day_number property.  For such a simple object is not a good idea.  An immutable object is better.
There are exactly 7 days of the week.  For a one-of-seven choice, the go-to data-storage structure is an Enum.  For example:
from enum import Enum

class DayOfWeek(Enum):
    Sunday = 1
    Monday = 2
    Tuesday = 3
    Wednesday = 4
    Thursday = 5
    Friday = 6
    Saturday = 7

    @property
    def weekend(self):
        return self in {DayOfWeek.Saturday, DayOfWeek.Sunday}

    @property
    def category(self):
        return 'Weekend' if self.weekend else 'Business day'

    @property
    def number(self):
        return self.value

Borrowing Reideerien's request_day_number() function you'd use the DayOfWeek enum like:
def main():
    day_no = request_day_number()
    day = DayOfWeek(day_no)

    print(f'Day {day.number} is {day.name}')
    print(f'{day.name} is a {day.category} day')

if __name__ == '__main__':
    main()