The problem was : For a given positive integer N, what is the maximum sum of distinct numbers such that the Least Common Multiple of all these numbers is N. for n=1) Only possible number is 1, so the maximum sum of distinct numbers is exactly 1.
for n= 2 The distinct numbers you can have are just 1 and 2, so the sum is 3. If we consider any other number greater than 2, then the least common multiple will be more than 2.
My interpretation of the question is that the set of numbers will be the set of all divisors of your given number $n$ (including $n$ itself).
In this case the answer depends on the prime factorization of $n$. Write $n = p_1^{q_1}p_2^{q_2}\dots p_i^{q_m}$, where the $p_i$ are distinct primes. The divisors will be of the form $p_1^{l_1}p_2^{l_2}\dots p_i^{l_m}$ for $0 \le l_i \le q_i$ for $i = 1, \ldots m$. The summation of all such divisors can be shown to be $\left(\frac{p_1^{q_1+1} -1}{p_1-1}\right)\left(\frac{p_2^{q_2+1} -1}{p_2-1}\right)\dots \left(\frac{p_m^{q_m+1} -1}{p_m-1}\right)$.
edit: a generalization of this function can be found at http://en.wikipedia.org/wiki/Divisor_function
Actually question is you have a given lcm say N we have to find the number of distinct pairs of numbers whose lcm is N.
For a given positive integer N, what is the maximum sum of distinct numbers such that the Least Common Multiple of all these numbers is N.
