Pronic number

A pronic number is a number that is the product of two consecutive integers, that is, a number of the form ${\displaystyle n(n+1)}$.^[1] The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers,^[2] or rectangular numbers;^[3] however, the term "rectangular number" has also been applied to the composite numbers.^[4][5]

The first 60 pronic numbers are:

0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450, 2550, 2652, 2756, 2862, 2970, 3080, 3192, 3306, 3422, 3540, 3660... (sequence A002378 in the OEIS).

Letting ${\displaystyle P_{n}}$ denote the pronic number ${\displaystyle n(n+1)}$, we have ${\displaystyle P_{{-}n}=P_{n{-}1}}$. Therefore, in discussing pronic numbers, we may assume that ${\displaystyle n\geq 0}$ without loss of generality, a convention that is adopted in the following sections.