Pronic Number

A pronic number, oblong number, rectangular 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:

0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462 …

These numbers are sometimes called oblong because they are analogous to polygonal numbers in this way:

1×2 2×3 3×4 4×5

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 and the only pronic Lucas number.

The number of off-diagonal entries in a square matrix is always a pronic number.

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.

Other articles related to "number, pronic number":

3000 (number) - Selected Numbers in The Range 3001–3999
... 3003 – triangular number, only number known to appear eight times in Pascal's triangle no number is known to appear more than eight times other than 1 ... of the cubes of the first ten integers, centered octagonal number 3045 – sum of the integers 196 to 210 and sum of the integers 211 to 3046 ... – centered heptagonal. 3249 – 57 2, centered octagonal number, member of a Ruth–Aaron pair with 3248 under second definition 3256 – centered heptagonal number 3266 – sum of first 41 ...
4000 (number) - Selected Numbers in The Range 4001–4999
... 4005 – triangular number 4007 – safe prime 4010 – magic constant of n×n normal magic square and n-queens problem for n = 20 ... of the first 45 primes 4030 – third weird number 4031 – sum of the cubes of the first six primes 4032 – pronic number 4033 – sixth super-Poulet number strong pseudoprime in base 2 4060 ... Also record number of wickets taken in first-class cricket by Wilfred Rhodes ...

Famous quotes containing the word number:

    There is not to be found, in all history, any miracle attested by a sufficient number of men, of such unquestioned good sense, education, and learning, as to secure us against all delusion in themselves ... beyond all suspicion of any design to deceive others ... and at the same time attesting facts, performed in such a public manner, and in so celebrated a part of the world, as to render the detection unavoidable.
    David Hume (1711–1776)