Upper bound for the fifth smallest prime divisor of a friend of 10

Establish an upper bound for the fifth smallest prime divisor q5 of any friend n of 10 in terms of the number of distinct prime divisors ω(n) by identifying positive integers A and B with A ≥ B such that q5 < p_{ceil(A · ω(n)/B)}, where p_k denotes the k-th prime number; equivalently, determine whether such A and B exist so that the approach used for q2, q3, and q4 in Theorems 1.1–1.3 extends to q5.

Background

The main results of the paper provide necessary upper bounds for the second, third, and fourth smallest prime divisors of any friend of 10, expressed in terms of the number of distinct prime divisors ω(n). These bounds are achieved via comparisons using properties of the abundancy index and estimates on prime gaps.

The authors explicitly state that they were unable to extend their method to the fifth smallest prime divisor q5, framing a concrete gap: finding positive integers A and B (with A ≥ B) that would allow the same technique to yield an inequality of the form q5 < p_{ceil(A ω(n)/B)}. This formulates a precise open problem for extending their approach.