Odd Number

An odd number is an integer of the form n=2k+1, where k is an integer. The odd numbers are therefore ..., -3, -1, 1, 3, 5, 7, ... (OEIS A005408), which are also the gnomonic numbers. Integers which are not odd are called even.

Odd numbers leave a remainder of 1 when divided by two, i.e., the congruence n=1 (mod 2) holds for odd n. The oddness of a number is called its parity, so an odd number has parity 1, while an even number has parity 0.

The generating function for the odd numbers is


The product of an even number and an odd number is always even, as can be seen by writing


which is divisible by 2 and hence is even.

See also

Connell Sequence, Consecutive Numbers, Even Number, Gnomonic Number, Nicomachus's Theorem, Odd Number Theorem, Odd Perfect Number, Odd Prime, Parity

Explore with Wolfram|Alpha


Commission on Mathematics of the College Entrance Examination Board. Informal Deduction in Algebra: Properties of Odd and Even Numbers. Princeton, NJ, 1959.Sloane, N. J. A. Sequence A005408/M2400 in "The On-Line Encyclopedia of Integer Sequences."Merzbach, U. C. and Boyer, C. B. A History of Mathematics, 3rd ed. New York: Wiley, p. 49, 1991.

Referenced on Wolfram|Alpha

Odd Number

Cite this as:

Weisstein, Eric W. "Odd Number." From MathWorld--A Wolfram Web Resource.

Subject classifications