Pronic number
From Wikipedia, the free encyclopedia
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. The first few pronic numbers (sequence A002378 in OEIS) are:
These numbers are sometimes called oblong because they are figurate 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 prionic 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.
[edit] Reference
- Dickson, L. E., History of the Theory of Numbers, Vol. 1: "Divisibility and Primality". New York: Dover, p. 357, 2005.
