There have been a lot of information published about bounds on lcm of polynomials and other types of sequences evaluated at consecutive naturals.


Moreover it’s known that $\operatorname{lcm}(1,2,3,\dotsc, n)>2^n$, and it behaves asymptotically as $e^n$.


However I wanted to ask two questions.


1. Does there exist an effective lower bound on cardinality of set $A\in\{1,2,3, \dotsc, n\}$ such that the lcm of the sequence is greater than $\phi^n$ where $\phi$ is the golden ratio?
2. Is there any rough asymptotic of the number of subsets $A$ such that the lcm of the members of sequence is greater than $\phi^n$?



Lower bound on the cardinality of set $A\in \{1,2,3,\dotsc,n\}$ with $\operatorname{lcm}(A)>\phi^n$ and asymptotic of number of such subsets
In the paper [The least common multiple of random sets of positive integers](https://arxiv.org/abs/1112.3013), Cilleruelo, Rué, Šarka, Zumalacárregui prove that if each subset of
$\{1,\dotsc,n\}$ is chosen with equal probability (equivalently each point $a\in \{1,\dotsc,n\}$ is included in $A$ with probability $\delta=1/2$) then for almost all sets $A\in \{1,\dotsc,n\}$ one has
$$
\operatorname{lcm}\{a: a \in A\}=2^{n(1+o(1))}.
$$
This would seem to indicate the answer to question 2 is asymptotically $2^n$.


