The logarithm of the Least common multiple, or $\log(\operatorname{LCM})$, of ${1, 2, \ldots, n} ={}$A003418 can be computed as the infinite series:
$$\log(A003418(n)) = \sum_{k \geq 1} \frac{T(n, k)}{k} - \frac{1}{k}$$
where the infinite matrix $T(n, k)$ = A309229 has the recurrence:
$$ \begin{split} T(n, 1) &= [n \geq 1] \; n;\\ \\ T(1, k) &= 1;\\ T(n, k) &= [n > k]\;T(n - k, k) + [n \leq k]\left(\sum_{i=0}^{n-1} T(n - 1, k - i) - \sum_{i=1}^{n - \color{Red}{c}} T(n, k - i)\right)\\ \end{split}$$
and where the constant $\color{Red}{c}$ in the upper summation limit of the last sum is equal to $\color{Red}{1}$, that is: $$\color{Red}{c=1}.$$
Perturb the recurrence by letting instead:
$$\color{Red}{c=2}$$
in the same recurrence and call the new table $t(n,k)$, which then becomes:
$$ \begin{split} t(n, 1) & = [n \geq 1] \; n;\\ \\ t(1, k) & = 1;\\ t(n, k) & = [n > k]\;t(n - k, k) + [n \leq k]\left(\sum_{i=0}^{n-1} t(n - 1, k - i) - \sum_{i=1}^{n - \color{Red}{2}} t(n, k - i)\right) \end{split}$$
Then compute the sum:
$$g(n,N)=\sum _{k=1}^{k=N} \frac{t(n,k)}{k} \tag{1}$$
Is the asymptotic of this sum: $$g(n,N) \approx h(n,N) = C \;\frac{\sqrt{n N} \log (n)}{\log (N)} \; ?$$
where $C$ is some constant.
Plots of the relation for $N = 5, 10, 25, 50, 100, 250, 500, 1000, 1250$
The plots were generated with Mathematica 14:
Clear[t, n, k, nn];
t[n_, 1] = If[n >= 1, n, 0];
t[1, k_] = 1;
t[n_, k_] :=
t[n, k] =
If[n > k, t[n - k, k], 0] +
If[n <= k,
Sum[t[n - 1, k - i], {i, 0, n - 1}] -
Sum[t[n, k - i], {i, 1, n - 2}], 0];
scale = {5, 10, 25, 50, 100, 250, 500, 1000,
1250}; (*all plots, computation time is about 5 to 10 minutes*)
scale = {5, 10, 25, 50, 100, 250}; (*fewer plots*)
Monitor[Do[
nn = scale[[j]];
g1 = ListLinePlot[
ParallelTable[Sum[t[n, k]/k, {k, 1, nn}], {n, 1, nn}],
PlotMarkers -> {Automatic, 5}];
g2 = ListLinePlot[
ParallelTable[(Sqrt[n*nn]*Log[n]/Log[nn]), {n, 1, nn}],
PlotStyle -> Red, PlotMarkers -> {Automatic, 5}];
g3 = ListLinePlot[
ParallelTable[
Sum[t[n, k]/k, {k, 1, nn}]/(Sqrt[n*nn]*Log[n]/Log[nn]), {n, 2,
nn}], PlotRange -> {0, 6}, PlotMarkers -> {Automatic, 5}];
Print[
GraphicsGrid[
Transpose[{{{Show[g1, g2], g3}[[1]]}, {{Show[g1, g2],
g3}[[2]]}}], ImageSize -> Large]], {j, 1, Length[scale]}];,
j]
