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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.

%I A086382 #12 Oct 14 2024 23:59:47
%S A086382 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 A086382 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 A086382 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 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.
%C A086382 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 A086382 Robert Israel, <a href="/A086382/b086382.txt">Table of n, a(n) for n = 2..10000</a>
%F A086382 a(3n)=1; a( A047235(n))=2
%p A086382 fibmod:= proc(k,m) uses LinearAlgebra:-Modular;
%p A086382   local M;
%p A086382   M:= Mod(m,<<0,1>|<1,1>>,integer[8]);
%p A086382   MatrixPower(m,M,k)[1,2]
%p A086382 end proc:
%p A086382 f:= proc(n) local k;
%p A086382    for k from 2 do if fibmod(k*n^2+1,k) <> fibmod(k+1,k) then return k-1 fi od
%p A086382 end proc:
%p A086382 map(f, [$2..100]); # _Robert Israel_, Oct 14 2024
%o A086382 (PARI) a(n)=if(n<0,0,m=1; while((fibonacci(m*n^2+1)-fibonacci(m+1))%m==0,m++); m-1)
%K A086382 nonn
%O A086382 2,1
%A A086382 _Benoit Cloitre_, Sep 06 2003