Finding the total number of set-bits from 1 to n

Question

Write an algorithm to find F(n) the number of bits set to 1, in all numbers from 1 to n for any given value of n.

Complexity should be O(log n)

Answer 1

If n= 2^k-1, then F(n)=k*(n+1)/2

For a general n, let m be the largest number such that m = 2^k-1 and m<=n. F(n) = F(m) + F(n-m-1) + (n-m).

Corner condition: F(0)=0 and F(-1)=0.