A pronic number, oblong 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:
These numbers are sometimes called oblong because they are figurate numbers in this way:
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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.
The number of off-diagonal entries in a square matrix is always a pronic number.
The value of the Möbius function μ(x) for any pronic number x = n (n + 1), in addition to being computable in the usual way, can also be calculated as
- μ(x) = μ(n) μ(n + 1).
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
- 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.
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