MathDB

6

Part of 1988 China National Olympiad

Problems(1)

China Mathematical Olympiad 1988 problem6

Source: China Mathematical Olympiad 1988 problem6

11/5/2013
Let nn (n≥3n\ge 3) be a natural number. Denote by f(n)f(n) the least natural number by which nn is not divisible (e.g. f(12)=5f(12)=5). If f(n)≥3f(n)\ge 3, we may have f(f(n))f(f(n)) in the same way. Similarly, if f(f(n))≥3f(f(n))\ge 3, we may have f(f(f(n)))f(f(f(n))), and so on. If f(f(…f⏟k times(n)… ))=2\underbrace{f(f(\dots f}_{k\text{ times}}(n)\dots ))=2, we call kk the “length” of nn (also we denote by lnl_n the “length” of nn). For arbitrary natural number nn (n≥3n\ge 3), find lnl_n with proof.
number theoryrelatively primenumber theory unsolved