On a problem regarding coefficients of cyclotomic polynomials

Let an(k) be the coefficient of tk in the nth cyclotomic polynomialΦn(t)=∏j=1gcd(j,n)=1n(t−e2πjin). Let M(k)=limx→∞1x∑n⩽xan(k) be the average of an(k), as introduced by Möller, and let fk=π26M(k)k∏q⩽kqprime(q+1). It was asked by Y. Gallot, P. Moree and H. Hommersom if the fk are integers for all k. In this paper, we prove that this is so. We further show that for any fixed natural number N, fk contains N as a factor for sufficiently large k.