Abstract: The arithmetic function f(n) which satisfies f(u2+kv2)=f2(u)+kf2(v) is further studied. It is proved that f(n) can be divided into three classes with k=6, and thus the conjecture of f(n) in the case of k=6 is verified. Whats more,some interesting results are presented. Some relations and distinction in the process of the proof with different values of parameter k are pointed out, which aims at providing some possible theoretical support for the complete proof of the conjecture.