Q: Is it true that for $n>3$, there exists $u$ and $v$ in $\mathbb N$ such that $$n=\sigma_1(u)+\sigma_1(v),$$ where $\sigma_1(k)$ is the sum of the positive integer divisors of $k$, $k\in \mathbb N$? I assume this is true and may seem to be connected with Goldbach conjecture but clearly should be a lot easier. For instance, if $n\ge 8$ is even, then assuming that Goldbach conjecture is true, $$n-2=p+q\implies\ n=p+1+q+1=\sigma(p)+\sigma(q)$$ for some primes $p$ and $q$. So, the answer to Q would be Yes if we assume Goldbach and $n$ even. There seems to be no simple way to do something similar for $n$ odd.
The numbers in the range of $\sigma_1(n)$ less than 100 are $$A=\{ 1, 3, 4, 6, 7, 8, 12, 13, 14, 15, 18, 20, 24, 28, 30, 31, 32, 36, 38, 39, 40, 42, 44, 48, 54, 56, 57, 60, 62, 63, 68, 72, 74, 78, 80, 84, 90, 91, 93, 96, 98\}.$$
The list of $a+b$ with $a,b\in A$, and then sorted is 2, 4, 4, 5, 5, 6, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20,... 20, 20
which makes the conjecture (Q true) very plausible. I wonder if this is already known. This is a question asked by Sourav Mandal on Research Gate.
As one can read below, it is likely that if $n\ge 7$, one can find $u$ a prime.