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

1,2

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.

Links

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A000045, A003601, A272412.

Sequence in context: A282809 A168156 A295403 * A276704 A103022 A086189

Adjacent sequences: A272437 A272438 A272439 * A272441 A272442 A272443

Keyword

nonn,easy

Author

Altug Alkan, Apr 29 2016

Status

approved