Let $M(n)$ be the the set of the multiplicative partitions of $n$, and let $D(n)$ be the set of the sum of the multiplicative partitions of the divisors of $n$.
eg $M(30)=\{\{30\},\{2,15\},\{3, 10\}, \{5, 6\}, \{2, 3, 5\}\}$
and $D(30)=\{30,17,13,11,10\}$
This can be normalised and plotted by pairing up the ordered set $\log D$ with $x=\{1,2,3\dots\}$, which in this case would yield the plot points $\{\{1,\log10\},\{2,\log11\},\{3,\log13\},\{4,\log17\},\{5,\log30\}\}$.
It appears that these points are normally distributed:
(plot of $\log F(6\ 469\ 693\ 230)$), which is fairly obvious intuitively. Is this an incarnation of the Erdős–Kac theorem? Likely proving this would be difficult?
Notes
Mathematica code for primorials (which are easiest to compute for large $n$):
<< Combinatorica`
primorial = 10;
a = Range[primorial];
b = Thread[a -> Prime[a]];
c = Apply[Times, SetPartitions[a] /. b, {2}];
ListLinePlot[Log[Sort[Map[Total[c[[#]]] &, Range[Length[c]]]]]]
General code for any $n$:
n = 1000;
α[1, r_] := α[1, r] = 1; α[n_, r_] := α[n, r] =
Module[{s, i}, s = Select[Divisors[n], 1 < # <= r &];
Sum[α[n/s[[i]], s[[i]]], {i, 1, Length[s]}]]; β[n_]:= α[n, n]; γ[lst_, p_] :=
Module[{t, i, j}, Union[Flatten[Table[t = lst[[i]]; t[[j]] = p*t[[j]];
Sort[t], {i, Length[lst]}, {j, Length[lst[[i]]]}], 1],
Table[Sort[Append[lst[[i]], p]], {i, Length[lst]}]]];
δ[n_] := Module[{i, j, p, e, lst = {{}}},{p, e} = Transpose[FactorInteger[n]];
Do[lst = γ[lst, p[[i]]], {i, Length[p]}, {j, e[[i]]}]; lst];
ListLinePlot[Log[Sort[Map[Total[δ[n][[#]]] &, Range[β[n]]]]],
InterpolationOrder -> 0]
Colossally abundant numbers also produce relatively smooth plots:
\[NumberSign][n_] := Times @@ Prime[Range[n]];
n = 2^3 3^1 \[NumberSign][5];