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 term "rectangular number" has also been applied to the composite numbers.[4][5]

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

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:

${\displaystyle \lfloor {\sqrt {n}}\rfloor \cdot \lceil {\sqrt {n}}\rceil =n}$

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 pronic numbers

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

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

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

${\displaystyle \sum _{i=1}^{n}{\frac {1}{i(i+1)}}={\frac {n}{n+1}}.}$

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

${\displaystyle \sum _{k=1}^{n}k(k+1)={\frac {n(n+1)(n+2)}{3}}=2T_{n}={\frac {n^{\overline {3}}}{3}}.}$

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] All 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.[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

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

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. "Plutarch, De Iside et Osiride, section 42". www.perseus.tufts.edu. Retrieved 16 April 2018.
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. (eds.), 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, archived from the original (PDF) on 2017-07-05, retrieved 2011-05-21.
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.