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 congruent 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 (x(1 + x))/(x - 1)^2 = x + 3x^2 + 5x^3 + 7x^4 + ....

Connell sequence | consecutive numbers | even number | gnomonic number | Nicomachus's theorem | odd number theorem | odd perfect number | odd prime | parity