A223488 Number of distinct residues in the Lucas sequence mod the n-th prime.

2, 3, 4, 7, 7, 12, 16, 12, 19, 10, 19, 28, 19, 33, 15, 44, 37, 28, 51, 44, 56, 49, 63, 24, 80, 35, 79, 33, 48, 40, 97, 82, 100, 33, 72, 37, 124, 123, 127, 124, 112, 62, 119, 144, 148, 16, 30, 169, 171, 80, 28, 149, 103, 157, 196, 85, 120, 169, 204, 27, 213, 212

Offset: 1

Author

Casey Mongoven, Mar 20 2013

Keywords

• nonn

Comments

The Lucas numbers mod n for any n are periodic; see A106291 for period lengths.

Examples

			The 5th prime number is 11. The Lucas sequence mod 11 is {2,1,3,4,7,0,7,7,3,10,2,1,3,...} - a periodic sequence. There are 7 distinct residues in this sequence, namely {0,1,2,3,4,7,10}. So a(5) = 7.
		

References

• V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA, 1969.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A137750.

Programs

Mathematica

pisano[n_] := Module[{a = {2, 1}, a0, k = 0, s}, If[n == 1, 1, a0 = a; Reap[While[k++; s = Mod[Plus @@ a, n]; Sow[s]; a[[1]] = a[[2]]; a[[2]] = s; a != a0]][[2, 1]]]]; Join[{2}, Table[u = Union[pisano[n]]; Length[u], {n, Prime[Range[2, 100]]}]] (* T. D. Noe, Mar 22 2013 *)