# 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).

## As figurate numbers

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]}

1×2 2×3 3×4 4×5

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 reciprocals

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]}

## Additional properties

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

- 1 2 3 Conway, J. H.; Guy, R. K. (1996),
*The Book of Numbers*, New York: Copernicus, Figure 2.15, p. 34. - 1 2 3 4 5 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. - 1 2 Ben-Menahem, Ari (2009),
*Historical Encyclopedia of Natural and Mathematical Sciences, Volume 1*, Springer reference, Springer-Verlag, p. 161, ISBN 9783540688310. - ↑ http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A2008.01.0238%3Asection%3D42
- ↑ Higgins, Peter Michael (2008),
*Number Story: From Counting to Cryptography*, Copernicus Books, p. 9, ISBN 9781848000018. - 1 2 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.