MathDB

Let

n

be a positive integer. Suppose for any

\mathcal{S} \subseteq \{1, 2, \cdots, n\}

f(\mathcal{S})

is the set containing all positive integers at most

n

that have an odd number of factors in

\mathcal{S}

. How many subsets of

\{1, 2, \cdots, n\}

can be turned into

\{1\}

after finitely many (possibly zero) applications of

f

Problem Statement