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Encyclopedia > 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 coprime.

  Results from FactBites:
Wikipedia search result (332 words)
Because 90 is the sum of its unitary divisors (excluding itself), it is a unitary perfect number, and because it is equal to the sum of a subset of its divisors, it is a semiperfect number.
It is a Perrin number, preceded in the sequence by 39, 51, 68.
The Saros number of the solar eclipse series which began on 134 BC September 28 and ended on 1345 March 4.
Pronic number - Wikipedia, the free encyclopedia (308 words)
The n-th pronic number is twice the n-th triangular number.
The number of off-diagonal entries in a square matrix is always a pronic number.
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.
  More results at FactBites »



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