A020995 Numbers k such that the sum of the digits of Fibonacci(k) is k.

0, 1, 5, 10, 31, 35, 62, 72, 175, 180, 216, 251, 252, 360, 494, 504, 540, 946, 1188, 2222

Offset

1,3

Comments

Since the number of digits in the k-th Fibonacci number ~ k*log_10(Golden Ratio), theoretically this sequence is infinite, but then the average density of those digits = ~ 0.208987. - Robert G. Wilson v

Robert Dawson of Saint Mary's University says it is likely that 2222 is the last term, as (assuming that the digits are equally distributed) the expected digit sum is ~ 0.9*k. - Stefan Steinerberger, Mar 12 2006 [Assuming that the average digit is (0+1+2+...+9)/10 = 9/2, the expected digit sum is ~ (9/2)*log_10((1+sqrt(5))/2)*k = 0.94044438...*k. - Jon E. Schoenfield, Aug 28 2022]

Bankoff's short paper lists the first seven terms. - T. D. Noe, Mar 19 2012

No more terms < 150000. - Manfred Scheucher, Aug 03 2015

If it exists, a(21) > 10^6. - Robert Price, May 26 2019

References

Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers, Prometheus Books, NY, 2007, page 209.

Links

Table of n, a(n) for n=1..20.

Leon Bankoff, A Fibonacci Curiosity, Fibonacci Quarterly 14, Feb. 1976, p. 17.

Pat Ballew, Fibonacci Digit Sums, Pat's Blog, Sunday, 5 August 2012.

Ron Knott, The Mathematical Magic of the Fibonacci Numbers: Digit Sums

Manfred Scheucher, Sage Script

David Terr, On the Sums of Digits of Fibonacci Numbers, Fibonacci Quarterly 34, Aug. 1996, pp. 349-355.

Example

Fibonacci(10) = 55 and 5+5 = 10.

Mathematica

Do[ If[ Apply[ Plus, IntegerDigits[ Fibonacci[n]]] == n, Print[n]], {n, 1, 10^5} ] (* Sven Simon *)
Do[ If[ Mod[ Fibonacci[n], 9] == Mod[n, 9], If[ Plus @@ IntegerDigits[ Fibonacci[n]] == n, Print[n]]], {n, 0, 10^6}] (* Robert G. Wilson v *)
Select[Range[0, 10^5], Plus @@ IntegerDigits[Fibonacci[ # ]] == # &] (* Ron Knott, Oct 30 2010 *)

Prog

(PARI) isok(n) = sumdigits(fibonacci(n)) == n; \\ Michel Marcus, Feb 18 2015

Crossrefs

Cf. A000045, A067515, A004090.

Sequence in context: A056422 A032296 A052648 * A174467 A005201 A221304

Adjacent sequences: A020992 A020993 A020994 * A020996 A020997 A020998

Keyword

nonn,base,more

Author

Sven Simon

Status

approved