In a search for integers $n$ whose divisors sum to a square (including $n$ itself among the divisors), it can be shown that 3 is the only odd prime whose divisor sum is square, and that no square of an odd prime has a square divisor sum. On the other hand, for odd composite $n$, square-free or not, 115 is the smallest of infinitely many odd composite $n$, none divisible by 3, whose divisor sums are square. This is shown in light of Dirichlet's theorem on the infinity of primes in any arithmetic progression $a + mb$, where $a$ and $b$ are co-prime.
If even $n = 2^m$, its divisor sum is not square for $m > 0$, but for $n$ containing any power of 2 and odd primes, square-free or not, 22 is the smallest of infinitely many even numbers whose divisor sums are square.
If $n$ is the cube of an odd prime $p$, it appears, with some help from triangle number theory, that $p^3$ is square (400) only for $p = 7$.
In all cases so far, the squares are even. But $n = p^4$, has square divisor sum for (and only for) $p = 3$. And the square is odd (121).
It is clear that the divisor sum must be even for odd powers of prime $p$, and odd for even powers of $p$. More generally, the divisor sum will be odd for $n$ containing $2^m$ (m > 0) and odd primes to even powers only. Odd divisor sums are a minority: the table through $n = 1000$ contains only 53 of them. 121 is the first square. And since 31 is the divisor sum of both 16 and 25, and for relatively prime integers, the sums are multiplicative, that is, the divisor sum of the product equals the product of the divisor sums of the factors, the divisor sum of 400 is a second odd square (961) in the table. 144 and 225 have divisor sum 403, but are not co-prime. Likewise for 242 and 196, with divisor sum 399. Going beyond the table, however, since 81 and 400 are relatively prime, the divisor sum of 81 x 400 = 32400 is $11^2$ x $31^2$ = $341^2$, a third odd square divisor sum.
Are there others? Or is it the case that 341, the smallest Fermat pseudoprime, is also the greatest odd number whose square is the divisor sum of an integer $n$? If so, is there a connection between odd square divisor sums and pseudoprimes?