Odd values of 
are 1, 1, 3, 5, 27, 89, 165, 585, ... (OEIS A051044),
and occur with ever decreasing frequency as 
becomes large (unlike 
, for which the fraction of odd values remains roughly 50%).
This follows from the pentagonal number theorem
which gives
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
(Gordon and Ono 1997), so 
is odd iff 
is of the form 
, i.e., 1, 5, 12, 22, 35, ... or 2, 7, 15, 26, 40,
....
The values of 
for which 
is prime are 3, 4, 5, 7,
22, 70, 100, 495, 1247, 2072, 320397, ... (OEIS A035359),
with no others for 
(Weisstein, May 6, 2000). These values correspond
to 2, 2, 3, 5, 89, 29927, 444793, 602644050950309, ... (OEIS A051005).
It is not known if 
is infinitely often prime, but Gordon and Ono (1997) proved
that it is "almost always" divisible by any given power of 2 (1997).
Gordon and Hughes (1981) showed that
 |
(4)
|
and
 |
(5)
|
where 
is an integer depending only on 
.
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