# Pronic number

A **pronic number** is a number which is the product of two consecutive integers, that is, a number of the form *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 "rectangular number" name 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).

If n is a pronic number, then the following is true:

## As figurate numbers[edit]

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 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.

## Sum of pronic numbers[edit]

The sum of the reciprocals of the pronic numbers (excluding 0) is a telescoping series that sums to 1:^{[6]}

The partial sum of the first n terms in this series is^{[6]}

The partial sum of the first n pronic numbers is twice the value of the nth tetrahedral number:

## Additional properties[edit]

The nth pronic number is the sum of the first n even integers.^{[2]}
It follows that all pronic numbers are even, and that 2 is the only prime pronic number. It is also the only pronic number in the Fibonacci sequence and the only pronic Lucas number.^{[7]}^{[8]}

The number of off-diagonal entries in a square matrix is always a pronic number.^{[9]}

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 e.g. 625 = 25^{2}, 1225 = 35^{2}. This is because

- .

## References[edit]

- ^
^{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}^{e}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.- ^
^{a}^{b}Frantz, Marc (2010), "The telescoping series in perspective", in Diefenderfer, Caren L.; Nelsen, Roger B.,*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.**^**McDaniel, Wayne L. (1998), "Pronic Fibonacci numbers" (PDF),*Fibonacci Quarterly*,**36**(1): 56–59, MR 1605341.**^**Rummel, Rudolf J. (1988),*Applied Factor Analysis*, Northwestern University Press, p. 319, ISBN 9780810108240.