A223489 - OEIS

%I #8 Aug 03 2014 14:01:41

%S 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,

%T 24,74,61,73,30,49,37,106,77,114,33,40,40,49,67,119,72,49,49,183,181,

%U 54,56,149,205,90,138,94,61,178,149,102,73,254,70,81,264,117,69

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

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

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

%H T. D. Noe, <a href="/A223489/b223489.txt">Table of n, a(n) for n = 1..1000</a>

%e 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.

%t 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 *)

%Y Cf. A137751.

%K nonn

%O 1,5

%A _Casey Mongoven_, Mar 20 2013