Algorithm to calculate the number of 1s for a range of numbers in binary

Question

So I just got back for the ACM Programing competition and did pretty well but there was one problem that not one team got.

The Problem.

Answer 1

I think it's a problem in Discrete mathematics,
assuming LOW is 0,
otherwise we can insert a function for summing numbers below LOW,
from numbers shown i understand the longest number will consist up to 60 binary digit at most

alg(HIGH,k)

l=len(HIGH)
sum=0;

for(i=0;i

all the numbers appear
non is listed twice
numbers_with_i_above is trivial
canreach with numbers up to 60 is easy
len is it length of a binary represention