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A223489 a(n) = number of missing residues in the Lucas sequence mod the n-th prime number. 1
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 (list; graph; refs; listen; history; text; internal format)
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: A046554 A010321 A046550 * A016521 A146880 A152236

Adjacent sequences:  A223486 A223487 A223488 * A223490 A223491 A223492

KEYWORD

nonn

AUTHOR

Casey Mongoven, Mar 20 2013

STATUS

approved

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Last modified July 31 13:25 EDT 2021. Contains 346373 sequences. (Running on oeis4.)