The odd part 
of a univariate polynomial 
is the sum of the monomials
containing odd powers of the variable.
Equivalently,
 |
It is an odd function, and 
is the even part of
. The same formula defines the odd
part of any univariate function with real
or complex values whose domain
is symmetric about 0.
In number theory, the odd part 
of a positive integer 
is defined by
 |
where 
is the greatest
dividing exponent of 2 in 
.
is therefore the product
of the prime factors of 
that are odd numbers, counted
with multiplicity. The values for 
, 2, ..., are 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, ... (OEIS
A000265).
