Perrin number

The Perrin numbers are a defined by the recurrence relation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_n := \begin{cases} 3 & \mbox{if } n = 0; \\ 0 & \mbox{if } n = 1; \\ 2 & \mbox{if } n = 2; \\ P_{n-2}+P_{n-3} & \mbox{if } n > 2. \\ \end{cases} }

The first few numbers of the sequence are: 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, ...

Properties

A special property of the sequence of Perrin numbers is, that if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\ } is a Prime number,than $p\$ divides Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_p\ } . The converse is false, because there exist composite numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\ } which divides Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_q\ } . Those numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\ } are called Perrin pseudoprimes. The first few Perrin pseudoprimes are: 271441, 904631, 16532714, 24658561, 27422714, ...