Let us consider a partition $1^{j_1} 2^{j_2} \cdots n^{j_n}$ of $n$ i.e. $j_1+2 j_2+\cdots + n j_n =n$. It iseasy to see that the sum $j_1+j_2+\cdots+j_n$ is just the length of the partition.
Question. Is there any combinatorial meaning or closed form of the two sums $$ j_1^2+2 j_2^2+\cdots + n j_n^2 $$ and $$ \sum_{k=1}^n \sum_{i=1}^{k-1} gcd(k,i) j_k j_i? $$
Just some initial observation about the first sum. $$j^2_1+2j^2_2+\dots+nj^2_n = j_1+2j_2+\dots+nj_n = n$$ exactly when the partition parts are distinct. Otherwise the sum with squares is giving some weighted measure of repeated parts.
$$\begin{array}{cc} \text{partition} & \text{sum with squares} \\ \hline (2,2) & 8 \\ (2,1,1) & 6 \\ (1,1,1,1) & 16 \\ \hline (3,1,1) & 7 \\ (2,2,1) & 9 \\ (2,1,1,1) & 11 \\ (1,1,1,1,1) & 25 \\ \hline \end{array}$$
The sequence of $j_1+2^2j_2+\dots+n^2j_n$ summed over all partitions of $n$ (that is, the sum of all parts squared in all partitions of $n$) is http://oeis.org/A066183, but the sequence of sum of $j^2_1+2j^2_2+\dots+nj^2_n$ summed over all partitions of $n$ isn't there. (I believe it starts $1,6,15,38,67$---comparing this to $n\cdot p(n)$, i.e., $1, 4, 9, 20, 35$, could say something about the repeated parts among all partitions of $n$...)