Let P(21)(n) denote the middle prime factor of n
(taking into account multiplicity). More generally, one can consider, for any
α∈(0,1), the α-positioned prime factor of n,
P(α)(n). It has previously been shown that loglogP(α)(n) has normal order αloglogx, and its values follow a
Gaussian distribution around this value. We extend this work by obtaining an
asymptotic formula for the count of n≤x for which P(α)(n)=p,
for primes p in a wide range up to x. We give several applications of these
results, including an exploration of the geometric mean of the middle prime
factors, for which we find that x1∑1<n≤xlogP(21)(n)∼A(logx)φ−1, where φ is the golden
ratio, and A is an explicit constant. Along the way, we obtain an extension
of Lichtman's recent work on the ``dissected'' Mertens' theorem sums
∑P+(n)≤yΩ(n)=kn1 for large values of
k