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.
As figurate numbers
The pronic numbers were studied as figurate numbers alongside the triangular numbers and square numbers in Aristotle's Metaphysics, and their discovery has been attributed much earlier to the Pythagoreans. As a kind of figurate number, the pronic numbers are sometimes called oblong because they are analogous to polygonal numbers in this way:
The nth pronic number is the sum of the first n even integers, and as such is twice the nth triangular number and n more than the nth square number, as given by the alternative formula n2 + n for pronic numbers. Hence the nth pronic number and the nth square number (the sum of the first n odd integers) form a superparticular ratio:
n ( n + 1 ) n 2 = n + 1 n {\displaystyle {\frac {n(n+1)}{n^{2}}}={\frac {n+1}{n}}}
Due to this ratio, the nth pronic number is at a radius of n and n + 1 from a perfect square, and the nth perfect square is at a radius of n from a pronic number. The nth pronic number is also the difference between the odd square (2n + 1)2 and the (n+1)st centered hexagonal number. Since the number of off-diagonal entries in a square matrix is twice a triangular number, it is a pronic number.
Sum of pronic numbers The partial sum of the first n positive pronic numbers is twice the value of the nth tetrahedral number:
∑ k = 1 n k ( k + 1 ) = n ( n + 1 ) ( n + 2 ) 3 = 2 T n {\displaystyle \sum _{k=1}^{n}k(k+1)={\frac {n(n+1)(n+2)}{3}}=2T_{n}} . The sum of the reciprocals of the positive pronic numbers (excluding 0) is a telescoping series that sums to 1:
∑ i = 1 ∞ 1 i ( i + 1 ) = 1 2 + 1 6 + 1 12 + 1 20 ⋯ = 1 {\displaystyle \sum _{i=1}^{\infty }{\frac {1}{i(i+1)}}={\frac {1}{2}}+{\frac {1}{6}}+{\frac {1}{12}}+{\frac {1}{20}}\cdots =1} . The partial sum of the first n terms in this series is
∑ i = 1 n 1 i ( i + 1 ) = n n + 1 {\displaystyle \sum _{i=1}^{n}{\frac {1}{i(i+1)}}={\frac {n}{n+1}}} . The alternating sum of the reciprocals of the positive pronic numbers (excluding 0) is a convergent series:
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