A086382 - OEIS

%I #12 Oct 14 2024 23:59:47

%S 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,

%T 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,

%U 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

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

%C 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

%H Robert Israel, <a href="/A086382/b086382.txt">Table of n, a(n) for n = 2..10000</a>

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

%p fibmod:= proc(k,m) uses LinearAlgebra:-Modular;

%p   local M;

%p   M:= Mod(m,<<0,1>|<1,1>>,integer[8]);

%p   MatrixPower(m,M,k)[1,2]

%p end proc:

%p f:= proc(n) local k;

%p    for k from 2 do if fibmod(k*n^2+1,k) <> fibmod(k+1,k) then return k-1 fi od

%p end proc:

%p map(f, [$2..100]); # _Robert Israel_, Oct 14 2024

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

%K nonn

%O 2,1

%A _Benoit Cloitre_, Sep 06 2003