Pronic number

[1] The study of these numbers dates back to Aristotle.

without loss of generality, a convention that is adopted in the following sections.

Hence the nth pronic number and the nth square number (the sum of the first n odd integers) form a superparticular ratio: Due to this ratio, the nth pronic number is at a radius of n and n + 1 from a perfect square, and the nth perfect square is at a radius of n from a pronic number.

The nth pronic number is also the difference between the odd square (2n + 1)2 and the (n+1)st centered hexagonal number.

[6] The partial sum of the first n positive pronic numbers is twice the value of the nth tetrahedral number: The sum of the reciprocals of the positive pronic numbers (excluding 0) is a telescoping series that sums to 1:[7] The partial sum of the first n terms in this series is[7] The alternating sum of the reciprocals of the positive pronic numbers (excluding 0) is a convergent series: Pronic numbers are even, and 2 is the only prime pronic number.

It is unique, since Another consequence of this chain of inequalities is the following property.

If m is a pronic number, then the following holds: 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.

This is so because The difference between two consecutive unit fractions is the reciprocal of a pronic number:[10]