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Two consecutive terms in the sequence are squares

Source: Bulgarian National Olympiad 2012 Problem 1

May 21, 2012

number theory proposednumber theory

Problem Statement

The sequence

a_1,a_2,a_3\ldots

, consisting of natural numbers, is defined by the rule:

a_{n+1}=a_{n}+2t(n)

for every natural number

n

, where

t(n)

is the number of the different divisors of

n

(including

1

and

n

). Is it possible that two consecutive members of the sequence are squares of natural numbers?