Pronic number

Demonstration, with Cuisenaire rods, of pronic numbers n=1, n=2, and n=3 (2, 6, and 12).

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

n(n + 1) = n² + n

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 n² + n for pronic numbers. The nth pronic number is also the difference between the odd square (2n + 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]

$1 = \frac{1}{2}+\frac{1}{6}+\frac{1}{12}\cdots=\sum_{i=1}^{\infty} \frac{1}{i(i+1)}.$

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

$\sum_{i=1}^{n} \frac{1}{i(i+1)} =\frac{n}{n+1}.$

Additional properties

First four pronic numbers as sums of the first n even numbers.

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 = 25², 1225 = 35². This is because

$(10n+5)^{2}=100n^{2}+100n+25=100n(n+1)+25\,$.

References

1. Conway, J. H.; Guy, R. K. (1996), The Book of Numbers, New York: Copernicus, Figure 2.15, p. 34.
2. 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.
3. Ben-Menahem, Ari (2009), Historical Encyclopedia of Natural and Mathematical Sciences, Volume 1, Springer reference, Springer-Verlag, p. 161, ISBN 9783540688310.
4. http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A2008.01.0238%3Asection%3D42
5. Higgins, Peter Michael (2008), Number Story: From Counting to Cryptography, Copernicus Books, p. 9, ISBN 9781848000018.
6. 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.
7. McDaniel, Wayne L. (1998), "Pronic Lucas numbers" (PDF), Fibonacci Quarterly, 36 (1): 60–62, MR 1605345.
8. McDaniel, Wayne L. (1998), "Pronic Fibonacci numbers" (PDF), Fibonacci Quarterly, 36 (1): 56–59, MR 1605341.
9. Rummel, Rudolf J. (1988), Applied Factor Analysis, Northwestern University Press, p. 319, ISBN 9780810108240.