This question is related to this post:Possible divisors of $s(2s+1)$. I have some follow up questions which should be a new post.
I write $\psi(s) = s(2s+1)$. We showed that for every prime $s$ that $4\mid d(\psi(s)$. I falsey conjectured that $\psi(s)$ could only take on the values $4,8,12,16,20$ and $24$. Take for example the prime $797161$ then $d(\psi(797161) =28$. And so this lead me to ask the following questions:
If $D$ is the set of all possible divisors of $\psi(s)$ does $D$ include every multiple of $4$? Equivalently if $D_{4m} = \{s|d(\psi(s) =4m\}$ is $D_{4m}$ empty for some integers $m$
Let $T$ be the table indexed in order column-wise with values from $D_{4m}$ then
$$ T = \begin{matrix} 2 & 7 & 31 & 67 & \ldots\\ 3 & 13 & 37 & 97 & \ldots\\ 5 & 17 & 73 & 127 & \ldots\\ 11 & 19 & 103 & 199 & \ldots\\ 23 & 43 & 137 & 227 & \ldots\\ 29 & 47 & 139 & 229 & \ldots\\ 41 & 59 & 181 & 241 & \ldots\\ \vdots & \vdots & \vdots & \vdots & \ddots\\ \end{matrix} $$
Another way of looking at my question is asking:
Does every column of $T$ have an entry?
Trivially $\bigcup D_{4m} = \text{PRIMES}$ - See Martin's comment below. In particular $T$ has infinitely many entries.
The entries in the first column of $T$ are the Sophie Germain primes. None of the other columns of $T$ can be found in Sloane's database. None of the rows or diagonals are in Sloan's database either.
Presumably every $D_{4m}$ is nonempty, but I believe we can't prove it yet.
The "hardest" case is when $m$ is prime: we can only have $\psi(s)=4m$ if $d(2s+1)=2m$, and there are only two types of numbers with exactly $2m$ divisors when $m$ is prime:
However, consider the polynomial $f(n) = (n^{2m-1}-1)/2$. It would follow from Schinzel's "Hypothesis H" that there are infinitely many primes $q$ such that $f(q)$ is prime. Then you can just take $s=f(q)$ for some such $q$, in which case $\psi(s)=4m$. (Indeed, this heuristic works for any $m$, prime or not.)