MathDB

For any positive integer

m

, denote by

P(m)

the product of positive divisors of

m

(e.g

P(6)=36

). For every positive integer

n

define the sequence a_1(n)=n,\qquad a_{k+1}(n)=P(a_k(n)) (k=1,2,\dots,2016) Determine whether for every set

S\subset\{1,2,\dots,2017\}

, there exists a positive integer

n

such that the following condition is satisfied:
For every

k

with

1\leq k\leq 2017

, the number

a_k(n)

is a perfect square if and only if

k\in S

Problem Statement