Pronic number

Template:Short description A pronic number is a number that is the product of two consecutive integers, that is, a number of the form $n(n+1)$. The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers, or rectangular numbers; however, the term "rectangular number" has also been applied to the composite numbers.

The first 60 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, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450, 2550, 2652, 2756, 2862, 2970, 3080, 3192, 3306, 3422, 3540, 3660... Template:OEIS.

Letting $P_n$ denote the pronic number $n(n+1)$, we have $P_{-n}=P_{n-1}$. Therefore, in discussing pronic numbers, we may assume that $n\ge 0$ without loss of generality, a convention that is adopted in the following sections.

As figurate numbers

File:Illustration of Triangular Number T 4 Leading to a Rectangle.png

Twice a triangular number is a pronic number

The Template:Mvarth pronic number is Template:Mvar more than the Template:Mvarth square number

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

* *	* * * * * *	* * * * * * * * * * * *	* * * * * * * * * * * * * * * * * * * *
1 × 2	2 × 3	3 × 4	4 × 5

The Template:Mvarth pronic number is the sum of the first Template:Mvar even integers, and as such is twice the Template:Mvarth triangular number and Template:Mvar more than the Template:Mvarth square number, as given by the alternative formula Template:Math for pronic numbers. Hence the Template:Mvarth pronic number and the Template:Mvarth square number (the sum of the [[Square_number#Properties|first Template:Mvar odd integers]]) form a superparticular ratio:

$\frac{n(n+1)}{n^2}=\frac{n+1}{n}$

Due to this ratio, the Template:Mvarth pronic number is at a radius of Template:Mvar and Template:Mvar + 1 from a perfect square, and the Template:Mvarth perfect square is at a radius of Template:Mvar from a pronic number. The Template:Mvarth pronic number is also the difference between the odd square Template:Math and the Template:Mathst centered hexagonal number.

Since the number of off-diagonal entries in a square matrix is twice a triangular number, it is a pronic number.

Sum of pronic numbers

The partial sum of the first Template:Mvar positive pronic numbers is twice the value of the Template:Mvarth tetrahedral number:

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

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

$\sum _{i=1}^{\mathrm{\infty }}\frac{1}{i(i+1)}=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}\cdots =1$.

The partial sum of the first Template:Mvar terms in this series is

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

The alternating sum of the reciprocals of the positive pronic numbers (excluding 0) is a convergent series:

$\sum _{i=1}^{\mathrm{\infty }}\frac{(-1{)}^{i+1}}{i(i+1)}=\frac{1}{2}-\frac{1}{6}+\frac{1}{12}-\frac{1}{20}\cdots =\log (4)-1$.

Additional properties

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.

The arithmetic mean of two consecutive pronic numbers is a square number:

$\frac{n(n+1)+(n+1)(n+2)}{2}=(n+1{)}^2$

So there is a square between any two consecutive pronic numbers. It is unique, since

$n^2\le n(n+1)<(n+1{)}^2<(n+1)(n+2)<(n+2{)}^2.$

Another consequence of this chain of inequalities is the following property. If Template:Mvar is a pronic number, then the following holds:

$\lfloor \sqrt{m}\rfloor \cdot \lceil \sqrt{m}\rceil =m.$

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 Template:Mvar or Template:Math. Thus a pronic number is squarefree if and only if Template:Mvar and Template:Math are also squarefree. The number of distinct prime factors of a pronic number is the sum of the number of distinct prime factors of Template:Mvar and Template:Math.

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 = 25² and 1225 = 35². This is so because

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

The difference between two consecutive unit fractions is the reciprocal of a pronic number:

$\frac{1}{n}-\frac{1}{n+1}=\frac{(n+1)-n}{n(n+1)}=\frac{1}{n(n+1)}$