Am I a 'Redivosite' Number?
Haskell, 91 85 83 80 75 74 bytes

n#m=([n#(div m d+d+1)|d&lt;-[2..m-1],mod m d&lt;1]++[mod n m&lt;1&amp;&amp;m&lt;n])!!0
f x=x#x


Try it online!

f x=x#x                           -- call # with x for both parameters
n#m               
         |d&lt;-[2..m-1],mod m d&lt;1   -- for all divisors d of m
    [n#(div m d+d+1)           ]  -- make a list of recursive calls to #,
                                  -- but with m set to m/d+d+1
   ++ [mod n m&lt;1&amp;&amp;m&lt;n]            -- append the Redivosite-ness of n (m divides n,
                                  -- but is not equal to n)
                           !!0    -- pick the last element of the list
                                  -- -&gt; if there's no d, i.e. m is prime, the
                                  --    Redivosite value is picked, else the
                                  --    result of the call to # with the smallest d


Edit: -2 bytes thanks to @BMO, -3 bytes thanks to @H.PWiz and -5 -6  bytes thanks to @Ørjan Johansen