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 pronic numbers were studied as figurate numbers alongside the triangular numbers and square numbers in Aristotle's *Metaphysics*,^{[2]} and their discovery has been attributed much earlier to the Pythagoreans.^{[3]}
As a kind of figurate number, the pronic numbers are sometimes called *oblong*^{[2]} because they are analogous to polygonal numbers in this way:^{[1]}

The nth pronic number is the sum of the first n even integers, and as such is twice the nth triangular number^{[1]}^{[2]} and n more than the nth square number, as given by the alternative formula *n*^{2} + *n* for pronic numbers. The nth pronic number is also the difference between the odd square (2*n* + 1)^{2} and the (*n*+1)st centered hexagonal number.

Since the number of off-diagonal entries in a square matrix is twice a triangular number, it is a pronic number.^{[6]}

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 = 25^{2} and 1225 = 35^{2}. This is so because

- .

- ^
^{a}^{b}^{c}Conway, J. H.; Guy, R. K. (1996),*The Book of Numbers*, New York: Copernicus, Figure 2.15, p. 34. - ^
^{a}^{b}^{c}^{d}Knorr, Wilbur Richard (1975),*The evolution of the Euclidean elements*, Dordrecht-Boston, Mass.: D. Reidel Publishing Co., pp. 144–150, ISBN 90-277-0509-7, MR 0472300. - ^
^{a}^{b}Ben-Menahem, Ari (2009),*Historical Encyclopedia of Natural and Mathematical Sciences, Volume 1*, Springer reference, Springer-Verlag, p. 161, ISBN 9783540688310. **^**"Plutarch, De Iside et Osiride, section 42",*www.perseus.tufts.edu*, retrieved 16 April 2018**^**Higgins, Peter Michael (2008),*Number Story: From Counting to Cryptography*, Copernicus Books, p. 9, ISBN 9781848000018.**^**Rummel, Rudolf J. (1988),*Applied Factor Analysis*, Northwestern University Press, p. 319, ISBN 9780810108240.- ^
^{a}^{b}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. **^**McDaniel, Wayne L. (1998), "Pronic Lucas numbers" (PDF),*Fibonacci Quarterly*,**36**(1): 60–62, MR 1605345, archived from the original (PDF) on 2017-07-05, retrieved 2011-05-21.**^**McDaniel, Wayne L. (1998), "Pronic Fibonacci numbers" (PDF),*Fibonacci Quarterly*,**36**(1): 56–59, MR 1605341.

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