TOPICS
Odd Part
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).
See also
Even Function, Even Part, Greatest Dividing Exponent, Odd Function, Odd Power, PolynomialExplore with Wolfram|Alpha
References
"Problem H-81." Fib. Quart. 6, 52, 1968.Sloane, N. J. A. Sequence A000265/M2222 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Odd PartCite this as:
Weisstein, Eric W. "Odd Part." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OddPart.html