An odd power of a quantity 
is a power 
whose exponent 
is an odd integer. For a polynomial or power series,
the odd powers are therefore 
,
, 
, ....
The odd part of a polynomial is the sum of its monomials containing odd powers of the variable. An odd function analytic at the origin has a Maclaurin series containing only odd powers.
In number theory, an odd power is also a number of the form 
, where 
is a positive integer
and 
is an odd
integer. The first few distinct odd powers are 1, 8, 27, 32, 64, 125, 128, 216,
243, 343, 512, ... (OEIS A070265). Every odd
power greater than 1 in this sense is a perfect power.
The converse does not hold. For example, 
is a perfect power but not an odd power. The double
series of reciprocals of the odd powers that are congruent to 3 (mod 4), counted
with multiplicity, is
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