%I
%S 1,3,5,6,21,41,45,65,67,68,78,96,109,382,497,517,527,658,682,705,759,
%T 805,930,966,1155,1557,1973,3211,3653,4563,5167,5620,9037,10027,10117,
%U 13279,17353,28856,35174,35534,45459,56072,154555,175151,177721,181561,183181,184201,184421,184601,185466,226666
%N Numbers n such that the average of the positive divisors of n is a Fibonacci number.
%C 1, 3, 5 and 21 are Fibonacci numbers. Are there other Fibonacci numbers in this sequence?
%C For a similar question and related proof attempt see the paper in the links section of A272412.
%H Charles R Greathouse IV, <a href="/A272440/b272440.txt">Table of n, a(n) for n = 1..734</a>
%e 3 is a term because 3 is divisible by 1 and 3. Average of 3 and 1 is 2 that is a Fibonacci number.
%t s = Array[Fibonacci, {28}]; Select[Range@ Max@ s, MemberQ[s, Mean@ Divisors@ #] &] (* _Michael De Vlieger_, Apr 29 2016 *)
%o (PARI) isFibonacci(n)=my(k=n^2); k+=((k + 1) << 2); issquare(k)  (n > 0 && issquare(k8))
%o is(n)=my(f=factor(n),s=sigma(f),d=numdiv(f)); s%d==0 && isFibonacci(s/d) \\ _Charles R Greathouse IV_, May 02 2016
%Y Cf. A000045, A003601, A272412.
%K nonn,easy
%O 1,2
%A _Altug Alkan_, Apr 29 2016
