JBMO-SL 2014 N6

Vukasin, Dimitrije, Dusan, Stefan and Filip asked their professor to guess a three consecutive positive integer numbers after they had told him these (true) sentences:

Vukasin: "Sum of the digits of one of them is a prime number. Sum of the digits of some of the other two is an even perfect number ( $n$ is perfect if $\sigma(n)=2 n$ ). Sum of the digits of the remaining number is equal to the number of its positive divisors."

Dimitrije:"Each of these three numbers has no more than two digits 1 in its decimal representation."

Dusan:"If we add 11 to one of them, we obtain a square of an integer."

Stefan:"Each of them has exactly one prime divisor less then 10."

Filip:"The 3 numbers are square-free."

Their professor gave the correct answer. Which numbers did he say?