ternary system

[quaxel]

I`m not sure, if this is the right category, but...

For every positive integer i, a_i is the greatest divisor of i which is not
is divisible by 3. The sequence (s_m) is defined by s_m := a₁ +a₂ + . . . + a_m.
Proofs: The number s_m is divisible by 3 precisely if the number of ones in the representation of m in the ternary system is divisible by 3.
(For example s₅ = 1 + 2 + 1 + 4 + 5)

My approaches are:
1. A number is divisible by three if it is its cross sum.
2. A number in the ternary system looks like this: a3⁰ b3¹ c3² ..., so if the ones in the representation are divisible by three, there are either no ones or 3x ones.

I would be very grateful for ideas or suggestions for solutions.