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A111419 a(n) is the smallest positive integer for which Fibonacci(n + a(n)) == Fibonacci(n) (mod n).

%I A111419 #12 Dec 19 2017 01:55:49
%S A111419 1,2,2,6,5,15,2,9,6,10,1,12,2,9,10,24,2,24,1,5,6,7,2,12,25,15,18,48,1,
%T A111419 15,1,11,14,19,10,12,2,15,34,60,1,15,2,30,30,25,2,12,14,50,42,78,2,24,
%U A111419 10,24,30,13,1,60,1,27,18,96,10,120,2,36,6,25,1,12,2,39,50,18,6,39,1,35
%N A111419 a(n) is the smallest positive integer for which Fibonacci(n + a(n)) == Fibonacci(n) (mod n).
%C A111419 When a(n)=2, n is often prime. The exceptions (323, 377, 2834, ...) are in A069107.
%H A111419 Michael De Vlieger, <a href="/A111419/b111419.txt">Table of n, a(n) for n = 1..10000</a>
%e A111419 a(3) = 2 because Fibonacci(3+2) - Fibonacci(3) = 5 - 2 == 0 (mod 3) and 2 is the smallest integer for which this is true.
%t A111419 Array[Block[{k = 1}, While[Mod[Fibonacci[# + k], #] != Mod[Fibonacci@ #, #], k++]; k] &, 80] (* _Michael De Vlieger_, Dec 17 2017 *)
%o A111419 (MuPAD) for n from 1 to 100 do an := 0; repeat an := an+1; until (numlib::fibonacci(n+an)-numlib::fibonacci(n)) mod n = 0 end_repeat; print(an); end_for;
%o A111419 (PARI) a(n) = {my(k = 1); while(Mod(fibonacci(n + k), n) != Mod(fibonacci(n), n), k++); k;} \\ _Michel Marcus_, Dec 18 2017
%Y A111419 Cf. A002708.
%K A111419 nonn
%O A111419 1,2
%A A111419 _Stefan Steinerberger_, Nov 13 2005