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). The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers, or rectangular numbers; however, the "rectangular number" name has also been applied to the composite numbers.

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 OEIS).

As figurate numbers

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 nth pronic number is twice the nth triangular number and n more than the nth square number, as given by the alternative formula n² + n for these 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 adds to 1:

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

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

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

Additional properties

The nth pronic number is the sum of the first n even integers. 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.

The number of off-diagonal entries in a square matrix is always a pronic number.

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 its factors. 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

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