A137750 -id:A137750 - cp's OEIS frontend

A137751 a(n) = number of missing residues in the Fibonacci sequence mod the n-th prime number.

0, 0, 0, 0, 4, 4, 4, 7, 4, 19, 12, 8, 22, 10, 32, 16, 22, 36, 16, 27, 16, 30, 20, 72, 28, 66, 24, 74, 60, 80, 30, 49, 28, 106, 88, 114, 44, 40, 40, 36, 67, 119, 72, 44, 48, 183, 181, 54, 56, 149, 212, 90, 138, 94, 64, 178, 156, 102

Offset: 1

Casey Mongoven, Feb 10 2008

nonn

			The 5th prime number is 11. The Fibonacci sequence mod 11 is {0,1,1,2,3,5,8,2,10,1,0,1,...} - a periodic sequence. There are 4 residues which do not occur in this sequence, namely {4,6,7,9}. So a(5) = 4.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Casey Mongoven, Absent Residues Primes no. 1; electro-acoustic music created with this sequence.

Cf. A118965, A137750, A000045.

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

Original entry on oeis.org

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

Casey Mongoven, Mar 20 2013

nonn

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

			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.

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

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

Cf. A137750.

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 *)

cp's OEIS Frontend

A137751 a(n) = number of missing residues in the Fibonacci sequence mod the n-th prime number.

Views

Author

Keywords

Examples

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica