cp's OEIS Frontend

A302062 a(n) is the smallest nonnegative k such that n + k divides Fibonacci(n) + k, or -1 if no such k exists.

%I A302062 #18 Oct 26 2020 09:33:58
%S A302062 1,0,-1,-1,-1,0,-1,-1,5,16,5,2,0,7,19,2,955,3,1265,2062,51,2,27,80,0,
%T A302062 0,121341,15080,317755,1,8,27,867,122,5,2,0,312,13029339,6680,25,2,40,
%U A302062 39,35,2,4537,15,0,1,95,2,67,7,8,15,32264490467,58,489,3108,0,22,1,116,973,2,125,19,143
%N A302062 a(n) is the smallest nonnegative k such that n + k divides Fibonacci(n) + k, or -1 if no such k exists.
%C A302062 Numbers n > 7 such that a(n) = Fibonacci(n) - 2*n are 8, 9, 16, 26, ...
%C A302062 From _Robert Israel_, Apr 02 2018: (Start)
%C A302062 For n > 7, a(n) = d - n where d is the least divisor of Fibonacci(n)-n that is not less than n.
%C A302062 In particular a(n) = Fibonacci(n)-2*n if n > 7 is in A270821. (End)
%F A302062 a(A023172(n)) = 0.
%F A302062 a(n) = -1 for n is in {2, 3, 4, 6, 7}. Otherwise, 0 <= a(n) <= Fibonacci(n) - 2*n.
%e A302062 a(7) = -1 because there is no nonnegative k such that k + 7 divides k + 13.
%e A302062 a(8) = 5 because 5 + 8 = 13 divides 5 + Fibonacci(8) = 26 and 5 is the only nonnegative integer with this property.
%e A302062 a(11) = 2 because 2 + 11 = 13 divides 2 + Fibonacci(11) = 91 and 2 is the least nonnegative integer with this property.
%p A302062 f:= proc(n) local t, d;
%p A302062   t:= combinat:-fibonacci(n);
%p A302062   subs(infinity=-1,min(select(`>=`,numtheory:-divisors(t-n),n))-n)
%p A302062 end proc:
%p A302062 f(0):= 1: f(1):= 0: f(5):= 0:
%p A302062 map(f, [$0..100]); # _Robert Israel_, Apr 02 2018
%t A302062 a[n_] := Module[{t = Fibonacci[n]}, Min[Select[Divisors[t-n], # >= n&]-n] /. Infinity -> -1];
%t A302062 a[0] = 1; a[1] = 0; a[5] = 0;
%t A302062 a /@ Range[0, 100] (* _Jean-François Alcover_, Oct 26 2020, after _Robert Israel_ *)
%Y A302062 Cf. A023172, A270821.
%K A302062 sign
%O A302062 0,9
%A A302062 _Altug Alkan_, Mar 31 2018