A272440 Numbers n such that the average of the positive divisors of n is a Fibonacci number.

1, 3, 5, 6, 21, 41, 45, 65, 67, 68, 78, 96, 109, 382, 497, 517, 527, 658, 682, 705, 759, 805, 930, 966, 1155, 1557, 1973, 3211, 3653, 4563, 5167, 5620, 9037, 10027, 10117, 13279, 17353, 28856, 35174, 35534, 45459, 56072, 154555, 175151, 177721, 181561, 183181, 184201, 184421, 184601, 185466, 226666

Offset: 1

Author

Altug Alkan, Apr 29 2016

Keywords

• nonn
• easy

Comments

1, 3, 5 and 21 are Fibonacci numbers. Are there other Fibonacci numbers in this sequence?

For a similar question and related proof attempt see the paper in the links section of A272412.

Examples

			3 is a term because 3 is divisible by 1 and 3. Average of 3 and 1 is 2 that is a Fibonacci number.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..734

Crossrefs

Cf. A000045, A003601, A272412.

Programs

Mathematica

s = Array[Fibonacci, {28}]; Select[Range@ Max@ s, MemberQ[s, Mean@ Divisors@ #] &] (* Michael De Vlieger, Apr 29 2016 *)

PARI

isFibonacci(n)=my(k=n^2); k+=((k + 1) << 2); issquare(k) || (n > 0 && issquare(k-8))
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