A086382 k divides F(k*n^2+1)-F(k+1) for 1<=k<=a(n) where F(k) is the k-th Fibonacci number.

2, 1, 2, 10, 1, 10, 2, 1, 2, 12, 1, 10, 2, 1, 2, 10, 1, 12, 2, 1, 2, 10, 1, 10, 2, 1, 2, 16, 1, 12, 2, 1, 2, 10, 1, 10, 2, 1, 2, 16, 1, 10, 2, 1, 2, 10, 1, 12, 2, 1, 2, 10, 1, 10, 2, 1, 2, 12, 1, 12, 2, 1, 2, 10, 1, 10, 2, 1, 2, 36, 1, 10, 2, 1, 2, 10, 1, 12, 2, 1, 2, 10, 1, 10, 2, 1, 2, 12, 1, 12

Offset: 2

Author

Benoit Cloitre, Sep 06 2003

Keywords

• nonn

Comments

Record values: a(2) = 2, a(5) = 10, a(11) = 12, a(29) = 16, a(71) = 36, a(3079) = 58. The next record a(n), if any has n > 10^5. - Robert Israel, Oct 14 2024

Links

• Robert Israel, Table of n, a(n) for n = 2..10000

Programs

Maple

fibmod:= proc(k,m) uses LinearAlgebra:-Modular;
  local M;
  M:= Mod(m,<<0,1>|<1,1>>,integer[8]);
  MatrixPower(m,M,k)[1,2]
end proc:
f:= proc(n) local k;
   for k from 2 do if fibmod(k*n^2+1,k) <> fibmod(k+1,k) then return k-1 fi od
end proc:
map(f, [$2..100]); # Robert Israel, Oct 14 2024

PARI

a(n)=if(n<0,0,m=1; while((fibonacci(m*n^2+1)-fibonacci(m+1))%m==0,m++); m-1)

Formula

a(3n)=1; a( A047235(n))=2