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τ(φ(n))=φ(τ(n)) where n has exactly two prime divisors

Source: Bulgaria MO 2011

May 30, 2011

number theoryrelatively primenumber theory proposed

Problem Statement

For each natural number

a

we denote

\tau (a)

and

\phi (a)

the number of natural numbers dividing

a

and the number of natural numbers less than

a

that are relatively prime to

a

. Find all natural numbers

n

for which

n

has exactly two different prime divisors and

n

satisfies

\tau (\phi (n))=\phi (\tau (n))