Am I a 'Redivosite' Number? Redivosite is a portmanteau word invented for the sole purpose of this challenge. It's a mix of Reduction, Division and Composite.

Definition

Given an integer N > 6:


If N is prime, N is not a Redivosite Number.
If N is composite:


repeatedly compute N' = N / d + d + 1 until N' is prime, where d is the smallest divisor of N greater than 1
N is a Redivosite Number if and only if the final value of N' is a divisor of N



Below are the 100 first Redivosite Numbers (no OEIS entry at the time of posting):

14,42,44,49,66,70,143,153,168,169,176,195,204,260,287,294,322,350,414,462,518,553,572,575,592,629,651,702,726,735,775,806,850,869,889,891,913,950,1014,1023,1027,1071,1118,1173,1177,1197,1221,1235,1254,1260,1302,1364,1403,1430,1441,1554,1598,1610,1615,1628,1650,1673,1683,1687,1690,1703,1710,1736,1771,1840,1957,1974,2046,2067,2139,2196,2231,2254,2257,2288,2310,2318,2353,2392,2409,2432,2480,2522,2544,2635,2640,2650,2652,2684,2717,2758,2760,2784,2822,2835


Examples


N = 13: 13 is prime, so 13 is not a Redivosite Number
N = 32: 32 / 2 + 3 = 19; 19 is not a divisor or 32, so 32 is not a Redivosite Number
N = 260: 260 / 2 + 3 = 133, 133 / 7 + 8 = 27, 27 / 3 + 4 = 13; 13 is a divisor or 260, so 260 is a Redivosite Number


Your task


Given an integer N, return a truthy value if it's a Redivosite Number or a falsy value otherwise. (You may also output any two distinct values, as long as they're consistent.)
The input is guaranteed to be larger than 6.
This is code-golf, so the shortest answer in bytes wins!
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