Pronic number

A pronic number, oblong number, rectangular number or heteromecic number, is a number which is the product of two consecutive integers, that is, n (n + 1). The n-th pronic number is twice the n-th triangular number and n more than the n-th square number. The first few pronic numbers (sequence A002378 in OEIS) are:

0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462 …

These numbers are sometimes called oblong because they are analogous to polygonal numbers in this way:

1×2	2×3	3×4	4×5

Pronic numbers can also be expressed as n² + n. The n-th pronic number is the sum of the first n even integers, as well as the difference between (2n − 1)² and the n-th centered hexagonal number.

All pronic numbers are even, therefore 2 is the only prime pronic number. It is also the only pronic number in the Fibonacci sequence and the only pronic Lucas number.^[1][2]

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. 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.

References

1. ^ Wayne L. McDaniel, "Pronic Lucas Numbers", The Fibonacci Quarterly, vol.36, iss.1, pp.60-62, 1998.
2. ^ Wayne L. McDaniel, "Pronic Fibonacci Numbers", The Fibonacci Quarterly, vol.36, iss.1, pp.56-59, 1998.

• Conway, J. H.; Guy, R. K. (1996), The Book of Numbers, New York: Copernicus, pp. 33–34 .
• Dickson, L. E. (2005), "Divisibility and Primality", History of the Theory of Numbers, 1, New York: Dover, p. 357 .