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LSpice
LSpice
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Can one show combinatorially how lcm$\operatorname{lcm}(1..n, \dotsc, n)$ grows?

Let us write $M(n)$ for lcm$(1,...,n)$$\operatorname{lcm}(1,\dotsc,n)$ for $n$ a positive integer. Asymptotically $M(n)$ tends toward $e^n$. This result uses analytic number theory. (Lcm is least common multiple, here of the first $n$ positive integers, and its logarithm is a Chebyshev function.)

It also seems that $2^n \lt M(n) \lt 3^n$ for $n \gt 6$ and even $2^n \leq M(n+1) \leq 3^n$ for $n \geq 0$. Are these relations true, and are there combinatorial proofs of either?

Additionally, do these inequalities appear in the combinatorial literature?

Gerhard "Interest In LCM Is Growing" Paseman, 2020.05.19.

Can one show combinatorially how lcm(1..n) grows?

Let us write $M(n)$ for lcm$(1,...,n)$ for $n$ a positive integer. Asymptotically $M(n)$ tends toward $e^n$. This result uses analytic number theory. (Lcm is least common multiple, here of the first $n$ positive integers, and its logarithm is a Chebyshev function.)

It also seems that $2^n \lt M(n) \lt 3^n$ for $n \gt 6$ and even $2^n \leq M(n+1) \leq 3^n$ for $n \geq 0$. Are these relations true, and are there combinatorial proofs of either?

Additionally, do these inequalities appear in the combinatorial literature?

Gerhard "Interest In LCM Is Growing" Paseman, 2020.05.19.

Can one show combinatorially how $\operatorname{lcm}(1, \dotsc, n)$ grows?

Let us write $M(n)$ for $\operatorname{lcm}(1,\dotsc,n)$ for $n$ a positive integer. Asymptotically $M(n)$ tends toward $e^n$. This result uses analytic number theory. (Lcm is least common multiple, here of the first $n$ positive integers, and its logarithm is a Chebyshev function.)

It also seems that $2^n \lt M(n) \lt 3^n$ for $n \gt 6$ and even $2^n \leq M(n+1) \leq 3^n$ for $n \geq 0$. Are these relations true, and are there combinatorial proofs of either?

Additionally, do these inequalities appear in the combinatorial literature?

Gerhard "Interest In LCM Is Growing" Paseman, 2020.05.19.

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Gerhard Paseman
Gerhard Paseman
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Can one show combinatorially how lcm(1..n) grows?

Let us write $M(n)$ for lcm$(1,...,n)$ for $n$ a positive integer. Asymptotically $M(n)$ tends toward $e^n$. This result uses analytic number theory. (Lcm is least common multiple, here of the first $n$ positive integers, and its logarithm is a Chebyshev function.)

It also seems that $2^n \lt M(n) \lt 3^n$ for $n \gt 6$ and even $2^n \leq M(n+1) \leq 3^n$ for $n \geq 0$. Are these relations true, and are there combinatorial proofs of either?

Additionally, do these inequalities appear in the combinatorial literature?

Gerhard "Interest In LCM Is Growing" Paseman, 2020.05.19.