Calculate the sum of digits of powers of 2: Let s_2(n) denote sum of digits in base-2 expansion of n

[safwane]

Let $s_2(n )$ denote the sum of the digits in the base-2 expansion of n.

Then my question is how one calculate $s_2(2^{m}),m≥1$. Here m is a positive integer.

 

[Dr.Peterson]

Let $s_2(n )$ denote the sum of the digits in the base-2 expansion of n.

Then my question is how one calculate $s_2(2^{m}),m≥1$. Here m is a positive integer.

What are you studying? What ideas do you have, from your context?

I would start by writing out the first few such numbers (m=1,2,3) and seeing what you find. Unless I'm misunderstanding the problem, you'll find it's a lot easier than it looks. Never just stare at a problem; do something!

 

[Steven G]

Let $s_2(n )$ denote the sum of the digits in the base-2 expansion of n.

Then my question is how one calculate $s_2(2^{m}),m≥1$. Here m is a positive integer.

What is the sum of the digits of 10^m in base 10.
How about the sum of the digits of 4^m in base 4.
How about base 7?
Dr Peterson is correct, that this a lot easier than it looks. In fact it is one of the easiest problems I have seen in a while. All you have to do is learn how to write powers of 2 in base 2. Hint: 2² = 10₂