Abstract:
In this note, we show that the set of n such that the arithmetic mean of the first n primes is an integer is of asymptotic density zero. We use the same method to show that the set of n such that the sum of die first n primes is a square is also of asymptotic density zero. We also prove that both the arithmetic mean of the first n primes as well as the square root of the sum of the first n primes are well distributed modulo 1.