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a_{n+1} = a_n +\tau (n), consective terms are perfect squares

Source: 2012 Balkan Shortlist BMO N1

April 5, 2020

number theorySequencePerfect SquaresconsecutivePerfect Square

Problem Statement

A sequence

(a_n)_{n=1}^{\infty}

of positive integers satisfies the condition

a_{n+1} = a_n +\tau (n)

for all positive integers

n

where

\tau (n)

is the number of positive integer divisors of

n

. Determine whether two consecutive terms of this sequence can be perfect squares.