Pronic number

A pronic number, or oblong number or heteromecic number, is a number which is the product of two consecutive nonnegative integers, that is, n(n + 1). Each pronic number for n is twice the triangular number for n. 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, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162

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

Clearly, 2 is the only prime pronic number. It is also the only pronic number in the Fibonacci sequence.

The value of the Möbius function, μ(x) for any pronic number, in addition to being computable in the usual way, can also be calculated multiplying μ(n) by μ(n + 1). If neither n nor its following neighbor are squarefree, then obviously neither will be the resulting pronic number. Perhaps not quite so obviously, if both n and its neighbor are numbers with an even number of prime factors, the resulting pronic number will also have an even number of prime factors. These observations follow from the properties that the Möbius function is multiplicative and that consecutive integers are relatively prime.