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%I #16 Jun 20 2013 14:41:29
%S 1,1,2,9,10,11,22,16,38,17,58,81,82,55,110,64,142,69,178,217,218,131,
%T 262,144,310,153,362,417,418,239,478,256,542,269,610,681,682,379,758,
%U 400,838,417,922,1009,1010,551,1102,576,1198,597,1298,1401,1402,755,1510
%N Minimum k such that (F(n) - 1) | sum_{i=n..n+k} F(i), where F(i) are Fibonacci numbers (A000045).
%C a(8*n) = 16*n^2, for n>0.
%C a(8*n + 2) = a(8*n + 1)/2-2, for n>=1.
%C a(8*n + 3) = 32*n^2 + 24*n + 2, for n>=0.
%C a(8*n + 5) = a(8*n + 4) + 1, for n>=0.
%C a(8*n + 7) = 2*a(8*n + 6), for n>=0.
%C Values of Sum_{i=n..n+k}{F(i)} / A000071(n) are listed in A226842.
%H Paolo P. Lava, <a href="/A226841/b226841.txt">Table of n, a(n) for n = 1..200</a>
%e The sum of first 10 Fibonacci numbers is 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143. We need to add at least 17 consecutive Fibonacci numbers, starting from F(11)=89, in order to have 89 + 144 + 233 + 377 + 610 + 987 + 1597 + 2584 + 4181 + 6765 + 10946 + 17711 + 28657 + 46368 + 75025 + 121393 + 196418 = 514085 and 514085 / 143 = 3595.
%p with(numtheory); with(combinat); ListA226841:= proc(q)
%p local n,a,b,k,p; a:=0;
%p for n from 1 to q do a:=a+fibonacci(n); b:=fibonacci(n+1); k:=1;
%p while not type(b/a,integer) do k:=k+1; b:=b+fibonacci(n+k); od; print(k); od; end: ListA226841(10^4);
%Y Cf. A000045, A000071.
%K nonn
%O 1,3
%A _Paolo P. Lava_, Jun 19 2013
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