imc

The first three values in the sum are equal to $1,$ the next five equal to $2,$ the next seven equal to $3,$ and the last one equal to $4.$ For example, since $2^2=4$ any square root of a number less than $4$ must be less than $2.$ Sum them all together to get $$3\cdot 1+5\cdot 2+7\cdot 3+1\cdot 4=3+10+21+4=\boxed{38}$$