Pronic
A pronic number is a number that is the product of two consecutive integers, that is, a number of the form
n
(
n
+
1
)
{\displaystyle n(n+1)}
. The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers, or rectangular numbers; however, the term "rectangular number" has also been applied to the composite numbers.
The first 60 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, 2256, 2352, 2450, 2550, 2652, 2756, 2862, 2970, 3080, 3192, 3306, 3422, 3540, 3660... (sequence A002378 in the OEIS).
Letting
P
n
{\displaystyle P_{n}}
denote the pronic number
n
(
n
+
1
)
{\displaystyle n(n+1)}
, we have
P
−
n
=
P
n
−
1
{\displaystyle P_{{-}n}=P_{n{-}1}}
. Therefore, in discussing pronic numbers, we may assume that
n
≥
0
{\displaystyle n\geq 0}
without loss of generality, a convention that is adopted in the following sections.
Smack Down
- 2022-10-21T00:00:00.000000Z
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