Which days can be start and end day of any century?

Question

Start with any number less than 20. If the number is even, halve it. If the number is odd, add one. Continue using the same rules writing each number one after the other. e.g. 17, 18, 9, 10, 5, 6, 3, 4, 2, 1.

Do they always end in 1? What is the longest list of numbers?

Answer 1

In the 1st century,
number of odd days= 5 odd days(Because the remainder after dividing 76*1+24*2 by 7 is 5). This implies that the last day of this century will be the 5th day of the week i.e. Friday.
In the 2nd century,
number of odd days=2*5 (twice of the extra days in one century)= 10 odd days. This implies the last day of this century will be Wednesday.
In the 3rd century,
number of odd days=15 days. The last day of this century will be Monday.
But the 4th century has no extra days since 400 is a leap year while 100. 200. and 300 weren't. and this gives 20+1 extra days. this amounts to 3 weeks with no days protruding out of a perfect week. The last day of this century is Sunday From 5th century onward, this cycle repeats.
The last days of a century can be Friday, Wednesday, Monday and Sunday.
Naturally, the first days of the following centuries will be Saturday, Thursday, Tuesday and Monday respective to the above list of last days.