A264935 Numbers k such that the average of the digits of the k-th Fibonacci number is greater than 5.

6, 11, 14, 16, 17, 20, 23, 24, 34, 38, 39, 42, 48, 49, 58, 66, 74, 77, 83, 87, 102, 104, 110, 120, 136, 145, 156, 158, 172, 177, 178, 183, 195, 201, 233, 235, 250, 256, 260, 273, 277, 282, 288, 293, 306, 319, 325, 329, 373, 389, 399, 415, 458, 512, 589, 609, 695, 862, 989, 1063

Offset

1,1

Comments

As k increases, it appears that the average of the digits of the k-th Fibonacci number approaches 9/2 (the same as would be expected with increasingly longer strings of random decimal digits).

a(60) = 1063 is almost certainly the last term in the sequence.

It seems nearly certain that there are only 11 Fibonacci numbers whose average digit is exactly 5; their indices are k = 5, 10, 35, 78, 97, 138, 184, 189, 300, 437, and 550.

Links

Table of n, a(n) for n=1..60.

Example

The first several terms and their corresponding Fibonacci numbers, number of digits D, digit sum S, and average digit values are as follows:
.
   k | Fibonacci(k) |  D |  S | avg. digit value
  ---+--------------+----+----+-----------------
   6 |            8 |  1 |  8 | 8.00000000000000
  11 |           89 |  2 | 17 | 8.50000000000000
  14 |          377 |  3 | 17 | 5.66666666666667
  16 |          987 |  3 | 24 | 8.00000000000000
  17 |         1597 |  4 | 22 | 5.50000000000000
  20 |         6765 |  4 | 24 | 6.00000000000000
  23 |        28657 |  5 | 28 | 5.60000000000000
  24 |        46368 |  5 | 27 | 5.40000000000000
  34 |      5702887 |  7 | 37 | 5.28571428571429
  38 |     39088169 |  8 | 44 | 5.50000000000000
  39 |     63245986 |  8 | 43 | 5.37500000000000
  42 |    267914296 |  9 | 46 | 5.11111111111111
  48 |   4807526976 | 10 | 54 | 5.40000000000000
  49 |   7778742049 | 10 | 55 | 5.50000000000000
  58 | 591286729879 | 12 | 73 | 6.08333333333333
.
(Fibonacci(58) is almost certainly the last Fibonacci number whose average digit exceeds 98/17 = 5.764705...)

Mathematica

Select[Range@ 1200, Mean@ IntegerDigits@ Fibonacci@ # > 5 &] (* Michael De Vlieger, Nov 28 2015 *)

Crossrefs

Cf. A000045.

Sequence in context: A163871 A201915 A237606 * A072540 A315374 A315375

Adjacent sequences: A264932 A264933 A264934 * A264936 A264937 A264938

Keyword

nonn,base

Author

Jon E. Schoenfield, Nov 28 2015

Status

approved