A223489 a(n) = number of missing residues in the Lucas sequence mod the n-th prime number.

0, 0, 1, 0, 4, 1, 1, 7, 4, 19, 12, 9, 22, 10, 32, 9, 22, 33, 16, 27, 17, 30, 20, 65, 17, 66, 24, 74, 61, 73, 30, 49, 37, 106, 77, 114, 33, 40, 40, 49, 67, 119, 72, 49, 49, 183, 181, 54, 56, 149, 205, 90, 138, 94, 61, 178, 149, 102, 73, 254, 70, 81, 264, 117, 69

Offset

1,5

Comments

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

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

Example

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 4 residues which do not occur in this sequence, namely {5,6,8,9}. So a(5) = 4.

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]]; mx = Max[u]; Length[Complement[Range[0, mx], u]], {n, Prime[Range[2, 100]]}]] (* T. D. Noe, Mar 22 2013 *)

Crossrefs

Cf. A137751.

Sequence in context: A046550 A363975 A355777 * A016521 A146880 A152236

Adjacent sequences: A223486 A223487 A223488 * A223490 A223491 A223492

Keyword

nonn

Author

Casey Mongoven, Mar 20 2013

Status

approved