Perrin number

Perrin, the sequence is a sequence of natural numbers, in which, similarly to the Fibonacci sequence, each link the sum of the previous elements (that is a defined sequence recursively ).

History

1876 ​​Édouard Lucas has been working on a series with the formation rule, but other starting values ​​possessed, as the Perrin sequence. 1899 R. Perrin has developed ideas of Lucas and from its formation rule with the initial values ​​P (0 ) = 3, P ( 1) = 0 and P (2 ) = 2 set up a series that has become known as the Perrin sequence.

Definition

Perrin sequence the elements are defined as follows:

This leads to the result:

It plays a role in graph theory, since P (n) is the number of maximum stable sets in a cyclic graph with n nodes.

Divisibility

In the following table, the first follower members 10 are shown, for which n is a divisor of P (n) is:

Interestingly, in this table all n, divide the P (n ) primes. In fact, it has proved that, when n is a prime number, n is the sequence P (n ) is divided. If you can draw the conclusion from this that, if n is the sequence P (n ) divides n is a prime number be? No, there are also composite n, divide the P (n). These composite n is called Perrinsche pseudo prime numbers ( sequence A013998 in OEIS ). The smallest Perrinsche pseudoprime is 271 441 = 5212th There are endless Perrinsche pseudoprimes.