Largest Mutually Coprime Subset of Initial Segment of Natural Numbers

Theorem

Let $n \in \N$ be a natural number.

Consider the set $\N_n$ defined as:

$\N_n = \closedint 1 n = \set {1, 2, \ldots n}$

Let $Q_n$ be the largest subset of $\N_n$ such that no element of $Q_n$ is the divisor of another element of $Q_n$.

Let $\map f n$ be the cardinality of $Q_n$.

Then for sufficiently large $n$:

$0 \cdotp 6725 \ldots \le \dfrac {\map f n} n \le 0 \cdotp 6736 \ldots$

Proof

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Sources

• 1977: Kenneth Lebensold: A divisibility problem (Studies in Applied Mathematics Vol. 56: pp. 291 – 294)
• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,6725 \ldots \leqslant d \leqslant 0,6736 \ldots$