On multiplicative functions which are additive on sums of primes

Let be a multiplicative function satisfying f(p (0)) not equal 0 for at least one prime number p (0), and let k a parts per thousand yen 2 be an integer. We show that if the function f satisfies f(p (1) + p (2) + . . . + p (k) ) = f(p (1)) + f(p (2)) + . . . + f(p (k) ) for any prime numbers p (1), p (2), . . . ,p (k) then f must be the identity f(n) = n for each . This result for k = 2 was established earlier by Spiro, whereas the case k = 3 was recently proved by Fang. In the proof of this result for k a parts per thousand yen 6 we use a recent result of Tao asserting that every odd number greater than 1 is the sum of at most five primes.