Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. For example, $\sigma(6)=1+2+3+6=12$ and $\sigma(28)=1+2+4+7+14+28=56$.
Denote the abundancy index $I$ of $x$ by $$I(x)=\frac{\sigma(x)}{x}.$$
If a positive integer $y$ is one of at least two solutions of $$I(y)=\frac{a}{b}$$ for a given rational number $a/b$, then $y$ is called a friendly number.
Here is a formal definition:
DEFINITION. Let $x$ and $y$ be distinct positive integers. If $x$ and $y$ satisfy the equation $I(x)=I(y)$ then $(x,y)$ is called a friendly pair. Each member of the pair is called a friendly number. (In other words, $x$ is a friend of $y$, and $y$ is also a friend of $x$.) A number which is not friendly is called a solitary number.
Checking the first few odd squares $N$ gives us $\gcd(N,\sigma(N))=1$, which by Greening's Theorem means that such $N$ are solitary. Since Greening's Theorem that
$\gcd(N,\sigma(N))=1$ implies that $N$ is solitary.
is sufficient (but not necessary), "it's easy to find odd squares for which $\gcd(n, \sigma(n))$ is not equal to $1$; e.g. if $n = {21}^2 = 441$ then $\gcd(n, \sigma(n)) = 3$. But $441$ is solitary." (Dean Hickerson, personal communication via e-mail, Mar. 12, 2011)
Here is my question:
Who is credited with the conjecture that all odd squares are solitary?
In an answer to a related MSE question (as hinted by Jeppe Stig Nielsen in the comments), it is conjectured that $9018009 = {3003}^2$ is not solitary.
It remains to find a friend of $9018009$, or to prove that none exists.