A pronic number is a number that is the product of two consecutive integers, that is, a number of the form [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 few 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 … (sequence A002378 in the OEIS).
Letting denote the pronic number we have Therefore, in discussing pronic numbers, we may assume that without loss of generality, a convention that is adopted in the following sections.
The partial sum of the first n positive pronic numbers is twice the value of the nth tetrahedral number:
The sum of the reciprocals of the positive pronic numbers (excluding 0) is a telescoping series that sums to 1:[7]
The partial sum of the first n terms in this series is[7]
The alternating sum of the reciprocals of the positive pronic numbers (excluding 0) is a convergent series:
Pronic numbers are even, and 2 is the only prime pronic number. It is also the only pronic number in the Fibonacci sequence and the only pronic Lucas number.[8][9]
The arithmetic mean of two consecutive pronic numbers is a square number:
So there is a square between any two consecutive pronic numbers. It is unique, since
Another consequence of this chain of inequalities is the following property. If m is a pronic number, then the following holds:
The fact that consecutive integers are coprime and that a pronic number is the product of two consecutive integers leads to a number of properties. Each distinct prime factor of a pronic number is present in only one of the factors n or n + 1. Thus a pronic number is squarefree if and only if n and n + 1 are also squarefree. The number of distinct prime factors of a pronic number is the sum of the number of distinct prime factors of n and n + 1.
If 25 is appended to the decimal representation of any pronic number, the result is a square number, the square of a number ending on 5; for example, 625 = 252 and 1225 = 352. This is so because
- .
Frantz, Marc (2010), "The telescoping series in perspective", in Diefenderfer, Caren L.; Nelsen, Roger B. (eds.), The Calculus Collection: A Resource for AP and Beyond, Classroom Resource Materials, Mathematical Association of America, pp. 467–468, ISBN 9780883857618.