A137751 -id:A137751 - cp's OEIS frontend

A137750 Number of distinct residues in the Fibonacci sequence mod the n-th prime.

2, 3, 5, 7, 7, 9, 13, 12, 19, 10, 19, 29, 19, 33, 15, 37, 37, 25, 51, 44, 57, 49, 63, 17, 69, 35, 79, 33, 49, 33, 97, 82, 109, 33, 61, 37, 113, 123, 127, 137, 112, 62, 119, 149, 149, 16, 30, 169, 171, 80, 21, 149, 103, 157, 193, 85

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 7 distinct residues in this sequence, namely {0,1,2,3,5,8,10}. So a(5) = 7.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Casey Mongoven, Unique Residues Primes no. 1; electro-acoustic music created with this sequence.

Cf. A066853, A137751, A000045, A001175.

With[{f=Fibonacci[Range[500]]},Table[Length[Union[Mod[f,Prime[n]]]],{n,60}]] (* Harvey P. Dale, Aug 15 2012 *)

a(n)=my(v=List([0,1]),p=prime(n));while(v[#v]||v[#v-1]!=1, listput(v,(v[#v]+v[#v-1])%p));#vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Apr 24 2012

a(n) = A066853(A000040(n)). - Max Alekseyev, Feb 08 2023

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

Original entry on oeis.org

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

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

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. A137751.

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 *)

cp's OEIS Frontend

A137750 Number of distinct residues in the Fibonacci sequence mod the n-th prime.

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