Perrin numbers

Let $P(0)=3, P(1)=0, P(2)=2$ and $$P(k) = P(k-2) + P(k-3.)$$ These numbers are called the Perrin numbers. They have the interesting property that $\mod{[P(k), k]} = 0$ in almost all cases when k is prime. ( Otherwise we would have found a true prime generator! ). In any case the property holds until $k=271441.$ Interesting, isn't it? See the table below for the first 40 Perrin numbers.

k	P[k]	Mod[P[k],k]	PrimeQ
2	2	0	True
3	3	0	True
4	2	2	False
5	5	0	True
6	5	5	False
7	7	0	True
8	10	2	False
9	12	3	False
10	17	7	False
11	22	0	True
12	29	5	False
13	39	0	True
14	51	9	False
15	68	8	False
16	90	10	False
17	119	0	True
18	158	14	False
19	209	0	True
20	277	17	False
21	367	10	False
22	486	2	False
23	644	0	True
24	853	13	False
25	1130	5	False
26	1497	15	False
27	1983	12	False
28	2627	23	False
29	3480	0	True
30	4610	20	False
31	6107	0	True
32	8090	26	False
33	10717	25	False
34	14197	19	False
35	18807	12	False
36	24914	2	False
37	33004	0	True
38	43721	21	False
39	57918	3	False
40	76725	5	False