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An equation of a function about prime divisors

Source: Turkey EGMO TST 2020 P2

February 6, 2020
functionnumber theory

Problem Statement

p(m)p(m) is the number of distinct prime divisors of a positive integer m>1m>1 and f(m)f(m) is the ⌊p(m)+12⌋\bigg \lfloor \frac{p(m)+1}{2}\bigg \rfloor th smallest prime divisor of mm. Find all positive integers nn satisfying the equation: f(n2+2)+f(n2+5)=2n−4f(n^2+2) + f(n^2+5) = 2n-4
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