A023175 Numbers k such that Fibonacci(k) == 3 (mod k).

1, 2, 4, 76, 124, 196, 244, 316, 374, 436, 556, 604, 724, 796, 844, 916, 964, 1084, 1324, 1396, 1516, 1636, 1684, 1756, 1768, 1804, 1996, 2164, 2284, 2404, 2476, 2524, 2644, 2764, 2836, 2956, 3004, 3076, 3244, 3316, 3436, 3676, 3964, 4036, 4084, 4156, 4204

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

Among the first 10000 terms, all are even except for 1 and 20017. - Robert Israel, Sep 22 2023

Contains k if k == 4 (mod A001175(k)). In particular, this is the case if k = 4 * p where p is a prime == 1 or 19 (mod 30). Most terms appear to be of this type, including all but 141 of the first 10000 terms. - Robert Israel, Sep 24 2023

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000045, A001175.

Programs

Maple

filter:= proc(n) uses LinearAlgebra:-Modular; local A;
  A:= Mod(n, Matrix([[0,1],[1,1]]),integer[]);
  MatrixPower(n,A,n)[1,2] - 3 mod n = 0
end proc:
filter(1):= true:
select(filte, [$1..10000]); # Robert Israel, Sep 22 2023

Mathematica

Select[Table[{p,Fibonacci[p]},{p,4500}],Mod[Last[#]-3,First[#]]==0&][[All,1]] (* Harvey P. Dale, Sep 22 2021 *)