N1

Part of 2012 Balkan MO Shortlist

Problems(1)

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

Source: 2012 Balkan Shortlist BMO N1

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

of positive integers satisfies the condition

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

for all positive integers

n

\tau (n)

is the number of positive integer divisors of

n

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

number theorySequencePerfect SquaresconsecutivePerfect Square