Top technology-crazy prefix and

\ [\ Sum _ {k = 1} ^ n f_k \]

Obviously, this item can be \ (O (n) \) and cannot be optimized.

But there are still faster algorithms in this universe ......

Order \ [g_n = \ sum _ {d | n} f_d, f_n = \ sum _ {k = 1} ^ {n} f_k \]

Because \ [\ sum _ {k = 1} ^ n G_k = \ sum _ {k = 1} ^ n {f_k \ lfloor \ frac NK \ rfloor} = \ sum _{ k = 1} ^ n f _ {\ lfloor \ frac NK \ rfloor} \]

\ [F_n = \ sum _ {k = 1} ^ n G_k-\ sum _ {k = 2} ^ n f _ {\ lfloor \ frac NK \ rfloor }\]

If \ (\ sum _ {k = 1} ^ n G_k \) can be obtained within \ (O (\ SQRT {n \), then \ (f (n) \) can be calculated in \ (O (N ^ \ frac 34 )\).

\ (\ Sum _ {k = 1} ^ n G_k \) for some special \ (G_k \) can be very fast, for example \ [\ sum _ {k = 1} ^ n \ sum _ {d | K} \ Mu (d) = 1 \] \ [\ sum _ {k = 1} ^ n \ sum _ {d | K} \ PHI (d) = \ sum _ {k = 1} ^ n k = \ frac {n (n + 1)} {2} \]

If \ (g_n \) can be quickly computed, then \ (n \ Leq n ^ \ frac 23 \) linear brute force calculation \ (f_n \) can be performed. Otherwise, recursive calculation is performed, \ (O (N ^ \ frac 23 )\).