Average Least Common Multiple

Problem Statement

The function $\operatorname{\mathbf{lcm}}(a,b)$ denotes the least common multiple of $a$ and $b$.
Let $A(n)$ be the average of the values of $\operatorname{lcm}(n,i)$ for $1 \le i \le n$.
E.g: $A(2)=(2+2)/2=2$ and $A(10)=(10+10+30+20+10+30+70+40+90+10)/10=32$.

Let $S(n)=\sum A(k)$ for $1 \le k \le n$.
$S(100)=122726$.

Find $S(2^N)$.

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